Probability
Black Line Masters
Draft
(NSSAL)
C. David Pilmer
© 2008
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The following are permitted to use and reproduce this resource for classroom purposes.
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The following are not permitted to use or reproduce this resource without the written
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Individuals, not including teachers or instructors, are permitted to use this resource for their own
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Introduction:
This Nova Scotia School for Adult Learning resource is designed for Level II Mathematics. It is
comprised of black line masters that instructors may wish to use with their learners when they
are learning about probability. These are individual activity sheets. Collectively the sheets do
not provide a comprehensive introduction to probability. Instructors should use other resources
in conjunction with these black line masters.
Table of Contents
Theoretical Probabilities (Part I)………………………………………………………………1
Answers………………………………………………………………………………. 6
Theoretical Probabilities (Part II)…………………………………………………………….. 7
Answers………………………………………………………………………………. 10
Experimental Probability (Part 1)…………………………………………………………….. 11
Experimental Probability (Part 2) ……………………………………………………………. 13
Answers………………………………………………………………………………. 22
What are the Chances? ………………………………………………………………………. 24
Answers………………………………………………………………………………. 27
Comparing Theoretical and Experimental Probability……………………………………….. 28
Answers………………………………………………………………………………. 33
Real World Probabilities……………………………………………………………………... 35
Answers………………………………………………………………………………. 38
The Lotto 649 Activity……………………………………………………………………… 40
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Theoretical Probability (Part 1)
Probability is the likelihood or chance of an event happening. When we
look at the flipping of a coin, there are two outcomes: head, tail. The
probability of obtaining a tail with a single flip of a coin can be found using
the following formula.
number of favourable outcomes
theoretical probability =
total number of possible outcomes
1
theoretical probability = , 0.50 or 50%
2
1
The probability of obtaining a tail when flipping a coin once is , 0.50 or 50%. This is referred
2
to as the theoretical probability. The theoretical probability is a measure of the likelihood of an
event, based on calculations.
Experimental probability is the measure of the likelihood of an event based on data from an
experiment. If you flipped a coin 10 times, you would expect to obtain tails 5 times but this is
not guaranteed. You might obtain tails 4 times. That would give you an experimental
probability of 0.4. It is important to note that even though the theoretical probability of getting
a tail was 0.5, the experimental probability turned out to be 0.4.
This activity sheet is concerned with theoretical probability. Remember we calculate theoretical
probability using the following formula.
number of favourable outcomes
theoretical probability =
total number of possible outcomes
Example 1:
What is the theoretical probability of obtaining a heart when you draw one card from a deck of
cards?
Answer:
Number of favourable outcomes: 13 (13 hearts in a deck of cards)
Total number of possible outcomes: 52 (52 cards in a deck)
13 1
Theoretical Probability =
, , 0.25 or 25%
52 4
Example 2:
What is the theoretical probability of obtaining a face card (King, Queen, Jack) when you draw
one card from a deck of cards?
Answer:
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Number of favourable outcomes: 12 (12 face cards in a deck of cards)
Total number of possible outcomes: 52 (52 cards in a deck)
12
Theoretical Probability =
, 0.23 or 23%
52
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Example 3:
What is the theoretical probability of obtaining an odd number when you spin the following
spinner once?
1
3
2
Answer:
Number of favourable outcomes: 2 (2 odd numbers: 1 and 3)
Total number of possible outcomes: 3 (3 numbers)
2
Theoretical Probability = , 0.67 or 67%
3
Example 4:
What is the theoretical probability of obtaining a number less than 3 when
you roll a six-sided die once?
Answer:
Number of favourable outcomes: 2
(2 numbers less than 3: 1 and 2)
Total number of possible outcomes: 6
(6 numbers of a six-sided die)
2 1
Theoretical Probability = , , 0.33 or 33%
6 3
Example 5:
What is the theoretical probability of obtaining a number greater than 6 when you roll a six-sided
die once?
Answer:
Number of favourable outcomes: 0 (none; there are no numbers greater than 6)
Total number of possible outcomes: 6 (6 numbers of a six-sided die)
0
Theoretical Probability = , 0 or 0%
6
Example 6:
What is the theoretical probability of obtaining a number between, and
including, 1 and 8 when you roll an eight-sided die once?
Answer:
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Number of favourable outcomes: 8
Total number of possible outcomes: 8
(8 numbers of a six-sided die)
8
Theoretical Probability = , 1 or 100%
8
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Questions:
Express all probabilities as a fraction, decimal and percent.
1. Fill in the blanks with the appropriate numbers.
(a) What is the theoretical probability of obtaining a 3, 4, or 5 with a single roll of a six-sided
die?
Number of favourable outcomes:
________
Total number of possible outcomes: ________
Theoretical Probability = ____________________
(b) What is the theoretical probability of obtaining a 2 with a single roll of an eight-sided
die?
Number of favourable outcomes:
________
Total number of possible outcomes: ________
Theoretical Probability = ____________________
(c) What is the theoretical probability of obtaining a number that is divisible by 3 with a
single roll of a twenty-sided die?
Number of favourable outcomes:
________
Total number of possible outcomes: ________
Theoretical Probability = ____________________
(d) What is the theoretical probability of obtaining a king when dealt one card from a deck of
cards?
Number of favourable outcomes:
________
Total number of possible outcomes: ________
Theoretical Probability = ____________________
(e) What is the theoretical probability of obtaining a king of hearts or diamonds when dealt
one card from a deck of cards?
Number of favourable outcomes:
________
Total number of possible outcomes: ________
Theoretical Probability = ____________________
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(f) What is the theoretical probability of obtaining an ace, 2, 3, or 4 when dealt one card from a
deck of cards?
Number of favourable outcomes:
________
Total number of possible outcomes: ________
Theoretical Probability = ____________________
2. Given the following spinner, find the probability of each event on a single spin.
9
1
2
8
7
3
6
4
5
(a) The number 5 is obtained.
___________________
(b) An even number is obtained.
___________________
(c) A number divisible by 3 is obtained.
___________________
(d) Any number other than 2 is obtained.
___________________
(e) A number greater than 9 is obtained.
___________________
(f) The number 3 or 7 is obtained.
___________________
(g) Any number between, and including, 1 and 9 is obtained.
___________________
3. The theoretical probability of winning grand prize in Lotto 649 is
1
.
13 983 816
(a) How many favourable outcomes are there to win the grand prize in Lotto 649?
(b) What is the total number of possible outcomes in Lotto 649?
(c) Knowing this information, would you take $20 and spend it on Lotto 649 tickets each
week? Explain.
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4. (a) If we have a fair six-sided die, can we say that each possible outcome (1, 2, 3, 4, 5, or 6)
is equally likely to occur?
(b) If we have a fair coin, can we say that each possible outcome (head or tail) is equally
likely to occur?
(c) If we had a spinner of the following configuration, can we
say that each possible outcome (A, B, C, or D) is equally
A
likely to occur?
D
B
C
(d) If you look back at questions 1, 2 and 3, did we focus on questions that dealt with
possible outcomes that are equally likely to occur or not?
5. What are your feelings about betting on games of chance (examples: poker, slot machines,
roulette)?
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Answers:
Please note that many of the decimal and percent answers had to be rounded off.
1. (F.O. - number of favourable outcomes, P.O. - total number of possible outcomes)
(a) F.O. = 3, P.O. = 6, Probability:
3 1
, , 0.50 or 50%
6 2
(b) F.O. = 1, P.O. = 8, Probability:
1
, 0.13 or 13%
8
(c) F.O. = 6, P.O. = 20, Probability:
6 3
,
, 0.30 or 30%
20 10
(d) F.O. = 4, P.O. = 52, Probability:
4 1
,
, 0.08 or 8%
52 13
(e) F.O. = 2, P.O. = 6, Probability:
2 1
,
, 0.04 or 4%
52 26
(f) F.O. = 16, P.O. = 52, Probability:
16 4
,
, 0.31 or 31%
52 13
1
, 0.11 or 11%
9
(b)
4
, 0.44 or 44%
9
(c)
3 1
, , 0.33 or 33%
9 3
(d)
8
, 0.89 or 89%
9
(e)
0
, 0 or 0%
9
(f)
2
, 0.22 or 22%
9
(g)
9
, 1 or 100%
9
2. (a)
3. (a) 1 favourable outcome
(b) 13 983 816 possible outcomes
(c) no
4. (a) yes
(c) no
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(b) yes
(d) equally likely
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Theoretical Probability (Part II)
If you completed the activity sheet called “Theoretical Probability (Part I)”, you learned to use
the following formula in situations where each outcome was equally likely to occur. For
example, with a six-sided die where each side is numbered 1 through 6, each outcome (1, 2, 3, 4,
5, or 6) is equally likely to occur.
theoretical probability =
number of favourable outcomes
total number of possible outcomes
This changes if the sides of the die are numbered
differently. Suppose three of the sides were numbered
as 1 and the remaining three sides were numbered 2
through 4. That means the chance of rolling a 1 is
three times greater than rolling a 2. The chance of
rolling a 1 is three times greater than rolling a 3. The
chance of rolling a 1 is three times greater than rolling
a 4. Each outcome (1, 2, 3, or 4) does not share the
same likelihood of occurring.
2
1
1
1
3
4
3 1
, , 0.50 or 50%
6 2
1
Theoretical Probability of Rolling a Two = , 0.17 or 17%
6
•
Theoretical Probability of Rolling a One =
•
Example 1:
You have a bag that contains 2 red marbles, 3 green marbles and 5 yellow marbles.
(a) If a single draw is made from the bag, what is the probability of obtaining a red marble?
(b) If a single draw is made from the bag, what is the probability of obtaining a green marble?
(c) If a single draw is made from the bag, what is the probability of obtaining a yellow marble?
(d) Does each outcome (red marble, green marble, yellow marble) share the same likelihood of
occurring? Explain.
(e) If a single draw is made from the bag, what is the probability of obtaining a green or red
marble?
(f) If a single draw is made from the bag, what is the probability of obtaining a blue marble?
Answers:
2 1
(a) , , 0.20 or 20%
10 5
(b)
3
, 0.30 or 30%
10
(c)
5 1
, , 0.50 or 50%
10 2
(d) Each outcome does not share the same likelihood of occurring because the probabilities
differ. The probability of obtaining a yellow marble (0.50) is higher than the probability
of obtaining a green marble (0.30). The probability of obtaining a green marble (0.30)
is higher than the probability of obtaining a red marble (0.20).
(e)
8 4
,
0.80 or 80%
10 5
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(f)
0
, 0 or 0%
10
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Questions:
Express all probabilities as a fraction, decimal and percent.
1. You have a six-sided die where two sides are numbered as 1, three sides are
numbered as 2, and one side is numbered as 3.
(a) What is the probability of rolling a 1 on a single roll?
____________
3
2
2
2
1
1
(b) What is the probability of rolling a 2 on a single roll?
____________
(c) What is the probability of rolling a 3 on a single roll?
____________
(d) What is the probability of rolling a 2 or 3 on a single roll?
____________
(e) Does each outcome (1, 2, or 3) share the same likelihood of occurring?
____________
(f) What is the probability of rolling a 1, 2 or 3 on a single roll?
____________
(g) What is the probability of rolling a 4 or 5 on a single roll?
____________
2. You have a six-sided die where each side is numbered 1 through 6. The
die is not a cube. It’s a rectangular-based prism (see diagram). Do all six
possible outcomes share the same likelihood of occurring? Explain.
3.
You have a bag that contains 6 red marbles, 12 green marbles and 2 blue marbles. You will
be making a single draw from the bag.
(a) What is the probability of obtaining a red marble?
____________
(b) What is the probability of obtaining a green marble?
____________
(c) What is the probability of obtaining a blue marble?
____________
(d) Does each outcome (red marble, green marble, blue marble) share
the same likelihood of occurring?
____________
(e) What is the probability of obtaining a purple marble?
____________
(f) What is the probability of obtaining a red or blue marble?
____________
(g) What is the probability of obtaining a red, green or blue marble?
____________
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4. Is the following statement true or false? Explain.
The probability of having a birthday in June is the same as the probability of having a
birthday in any other month.
5. You have a spinner of the following configuration.
B
A
C
(a) What is the probability of obtaining A on a single spin?
__________________
(b) What is the probability of obtaining B on a single spin?
__________________
(c) What is the probability of obtaining A or C on a single spin?
__________________
(d) What is the probability of obtaining D on a single spin?
__________________
(e) What is the probability of obtaining A, B, or C on a single spin?
__________________
6. You are taking random shots (no aiming) at the following
square dart board where each square has specific point values.
These values are marked on the board.
2
4
2
4
2
2
1
2
3
2
(a) What is the probability of obtaining a 1 on a single throw?
__________________
3
2
1
4
3
2
4
3
4
2
4
2
4
2
1
(b) What is the probability of obtaining a 2 on a single throw?
__________________
(c) What is the probability of obtaining a 4 on a single throw?
__________________
(d) What is the probability of obtaining a 1, 2, or 3 on a single throw?
__________________
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Answers:
Please note that many of the decimal and percent answers had to be rounded off.
2 1
, , 0.33, or 33%
6 3
1
(c) , 0.17, or 17%
6
1. (a)
(e) no
(f)
3 1
, , 0.50, or 50%
6 2
4 2
(d) , , 0.67, or 67%
6 3
6
(f)
, 1, or 100%
6
(b)
0
, 0, or 0%
6
2. No - This die is far more likely to rest on one of the four rectangular faces, rather than the
two square faces.
6 3
,
, 0.30, or 30%
20 10
2 1
(c)
,
, 0.10, or 10%
20 10
0
(e)
, 0, or 0%
20
20
(g)
, 1, or 100%
20
3. (a)
(b)
12 3
, , 0.60, or 60%
20 5
(d) no
(f)
8 2
, , 0.40, or 40%
20 5
4. No - Some months have 30 days, others have 31 days, and February as 28 days (29 on leap
year).
1
, 0.50, or 50%
2
3
(c) , 0.75, or 75%
4
(e) 1, or 100%
5. (a)
3
, 0.12, or 12%
25
7
(c)
, 0.28, or 28%
25
6. (a)
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(b)
1
, 0.25, or 25%
4
(d) 0, or 0%
11
, 0.44, or 44%
25
18
(d)
, 0.72, or 72%
25
(b)
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Experimental Probability (Part 1)
Probability is the likelihood or chance of an event happening. Experimental probabilities are
based on experiments where data is collected. For example, if you wanted to know Steve Nash’s
probability of making a three-point shot while playing college basketball, you could examine his
college record. He made 263 three-point shots out of 656 attempts while playing for Santa Clara
University. This data can be used to calculate the experimental probability.
number of times an outcome occurs
number of times the experiment is conducted
263
=
or 0.40 or 40%
656
Experimental Probability =
Example 1:
(a) Roll a six-sided die 12 times. Record the number of times you roll a 5. Based on this
experiment, calculate the experimental probability of rolling a 5 on a single roll of this die.
(b) Roll a six-sided die 24 times. Record the number of times you roll a 5. Based on this
experiment, calculate the experimental probability of rolling a 5 on a single roll of this die.
(c) Roll a six-sided die 36 times. Record the number of times you roll a 5. Based on this
experiment, calculate the experimental probability of rolling a 5 on a single roll of this die.
Answers: (Answers will vary from student to student.)
(a) When the die was rolled 12 times, I obtained a 5 only one time.
number of times an outcome occurs
Experimental Probability =
number of times the experiment is conducted
1
or 0.08 or 8%
=
12
(b) When the die was rolled 24 times, I obtained a 5 three times.
3
Experimental Probability =
or 0.13 or 13%
24
(c) When the die was rolled 36 times, I obtained a 5 seven times.
7
Experimental Probability =
or 0.19 or 19%
36
Note: If you rolled the die 1000 times or more, you would find that the experimental
probability of rolling a 5 on a single roll would approach the value 0.17 or 17%.
Questions:
1. (a) Take a coin and toss it 10 times. Record the number of times it comes up heads.
Calculate the experimental probability of obtaining a heads with this coin on a single flip.
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(b) Take a coin and toss it 40 times. Record the number of times it comes up heads.
Calculate the experimental probability of obtaining a heads with this coin on a single flip.
(c) Google search Shodor Interactive Activities Coin Toss. This particular online program
allows you to simulate multiple tosses of a coin. Enter 200 into the ‘Number of Tosses’
box and click on ‘Toss’em!’ The program will list the heads (H) and tails (T) you
obtained in your 200 tosses. If you select ‘Table’ from the ‘Display Results’, the number
of heads and the number of tails will be displayed. Use these results to calculate the
experimental probability of obtaining a heads on a single flip of a coin.
(d) Using the Shodor coin toss program, simulate 1000 tosses of a coin. Use these results to
calculate the experimental probability of obtaining a heads on a single flip of a coin.
(e) The theoretical probability of obtaining a head on a single toss of a coin is 0.5 or 50%.
Which of your experimental probabilities is closest to this theoretical probability?
2. Take a six-sided die and roll it 20 times. Record the number of times that you obtain a 1 or
2. Using the data you just collected, calculate the experimental probability of obtaining a 1
or 2 on a single roll of a six-sided die.
3. Using a small piece of corrugated cardboard (at least 8 cm by 8 cm), a
paper clip, and a thumbtack, construct a spinner that is divided into
four equal sections that are lettered A through D. The paper clip will
be the spinning arm and the thumbtack will hold the paper clip so that
it can spin around the center. Spin the spinner 20 times recording the
number of times it lands in section A. Use this data to calculate the
experimental probability of landing in section A with one spin of this
spinner.
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A
B
C
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Experimental Probability (Part 2)
Probability is the likelihood or chance of an event happening. Experimental probabilities are
based on experiments where data is collected. For example, if you wanted to know the
probability that a particular professional baseball player is going hit the ball the next time at bat,
then you would look at the player’s batting history. Suppose the player was at bat 278 times and
hit safely 102 times. This data can be used to calculate the experimental probability.
number of times an outcome occurs
number of times the experiment is conducted
102
=
or 0.37 or 37%
278
Experimental Probability =
We could interpret this to mean that this player has a 37% chance of a safe hit the next time at
bat. It could also mean that he has a 63% chance of not getting a safe hit the next time at bat.
Example 1:
Marcy wants to invent a board game for her children. She wants to use a die in the
game but decides to use a eight-sided die, rather than the traditional six-sided die.
Her die is a hexagonally-based prism (see diagram) where the numbers 1 through
6 are on the six rectangular faces, and the numbers 7 and 8 are on the two
hexagonal faces.
(a) Roll the die 80 times and complete a tally/frequency chart where you record the number of
times you roll each of the 8 numbers.
(b) What is the experimental probability that you will roll a 4 on a single roll?
(c) What is the experimental probability that you will roll a 2 on a single roll?
(d) What is the experimental probability that you will roll a 7 on a single roll?
(e) Which numbers have a greater likelihood of being rolled?
Answers:
(a) (Sample Data)
Tally
Roll a 1
Frequency
5
Roll a 2
8
Roll a 3
7
Roll a 4
9
Roll a 5
8
Roll a 6
4
Roll a 7
17
Roll an 8
22
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number of times an outcome occurs
number of times the experiment is conducted
9
Exp. Probability of Rolling a 4 =
or 0.11 or 11%
80
(b) Experimental Probability =
number of times an outcome occurs
number of times the experiment is conducted
8
Exp. Probability of Rolling a 2 =
or 0.10 or 10%
80
(c) Experimental Probability =
number of times an outcome occurs
number of times the experiment is conducted
17
Exp. Probability of Rolling a 7 =
or 0.21 or 21%
80
(d) Experimental Probability =
(e) The numbers 7 and 8 have the greatest likelihood of being rolled.
Example 2:
Marcus, Jason, and Hamid play basketball in a gentleman’s league. The three are trying to figure
out who is best at foul shots. Their foul shot data for the last few years is listed below. Use your
knowledge of probability to determine which one of the three players is the best at foul shots.
Marcus
Jason
Hamid
Number of Attempted
Foul Shots
32
27
29
Number of Successful
Foul Shots
19
14
21
Answer:
number of times an outcome occurs
number of times the experiment is conducted
19
=
or 0.59 or 59%
32
14
Jason:
Experimental Probability =
or 0.52 or 52%
27
21
Hamid:
Experimental Probability =
or 0.72 or 72%
29
Since Hamid has the highest probability (72%) of successfully completing foul shots, then he
is the best at foul shots.
Marcus:
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Experimental Probability =
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Example 3:
Jacob went to the Environment Canada website and was able to collect the following data
regarding the total rainfall in Halifax during the month of August from 1980 to 2007.
Year
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
Total Rainfall
(mm)
30.9
100.3
103.9
160.8
96.7
140.5
126.8
64.6
68.1
60.4
Year
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
Total Rainfall
(mm)
69.5
131.1
63.9
44.8
61.5
65.2
20.0
46.1
66.6
89.1
Year
2000
2001
2002
2003
2004
2005
2006
2007
Total Rainfall
(mm)
66.7
42.9
60.8
no data
127.6
24.7
68.0
195.5
(a) Create a tally/frequency chart where the classes are 0 - 25, 25 - 50, 50 - 75, 75 - 100, 100 125, 125 - 150, 150 - 175, and 175 - 200. For example, the total rainfall in August 1980
(30.9 mm) would fit in the class 25 - 50 because 30.9 mm is between 25 mm and 50 mm.
(b) What is the probability that the rainfall in August will be between 125 mm and 150 mm?
(c) What is the probability that the rainfall in August will be between 50 mm and 75 mm?
(d) What is the probability that the rainfall in August will be between 75 mm and 100 mm?
(e) What is the probability that the rainfall in August will be between 150 mm and 175 mm?
(f) How likely is it that Halifax will have between 150 mm and 175 mm of rainfall in August?
Answers:
(a) Class
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Tally
Frequency
0 - 25
2
25 - 50
4
50 - 75
11
75 - 100
2
100 - 125
2
125 - 150
4
150 - 175
1
175 - 200
1
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number of times an outcome occurs
number of times the experiment is conducted
4
=
or 0.15 or 15%
27
There is a 15% chance that the total rainfall in Halifax during August will be between
125 mm and 150 mm.
(b) Experimental Probability =
number of times an outcome occurs
number of times the experiment is conducted
11
=
or 0.41 or 41%
27
There is a 41% chance that the total rainfall in Halifax during August will be between 50
mm and 75 mm.
(c) Experimental Probability =
2
or 0.07 or 7%
27
There is a 7% chance that the total rainfall in Halifax during August will be between 75
mm and 100 mm.
(d) Experimental Probability =
1
or 0.04 or 4%
27
There is a 4% chance that the total rainfall in Halifax during August will be between 150
mm and 175 mm.
(e) Experimental Probability =
(f) It is very unlikely that there will be between 150 mm and 175 mm of rainfall in August
because the probability is so low (only 4%).
Questions:
Express all probabilities as a fraction, decimal and percent.
1. Andrea played 378 holes of golf this summer. She made par on 123 of those holes. What is
the probability that Andrea will make par on a hole?
2. Of 129 098 US college students who studied outside of the country, 32 237 students studied
in the United Kingdom. What is the probability that a US student studying outside of the
country will study in the United Kingdom?
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3. As of March 24, 2006, a total of 186 people worldwide had become ill due to the bird flu. Of
those, 22 people had gotten the flu in Thailand and 93 people had gotten it in Vietnam.
(a) Determine the probability that a person with the bird flu had gotten it in Thailand.
(b) Determine the probability that a person with the bird flu had gotten it in Vietnam.
4. In 2004, 2 104 661 males and 2 007 391 females were born in the US.
(a) How many births occurred in 2004?
________________
(b) What was the experimental probability that a US child born in 2004 was a male?
(c) What was the experimental probability that a US child born in 2004 was a female?
5. A new drug is being tested that is supposed to lower blood pressure. This drug is given 200
people and the results are listed below.
Number of People Whose Blood Pressure Lowered
Number of People Whose Blood Pressure Remained the Same
Number of People Whose Blood Pressure Got Higher
142
38
20
(a) What is the experimental probability that a person’s blood pressure would lower on this
new drug?
(b) What is the experimental probability that a person’s blood pressure would rise on this
new drug?
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(c) If you had high blood pressure, would you consider using this new drug? Explain.
6. Gavin worked in a department store that was having its annual sale. Every customer who
was buying an item received a Scratch and Save card at the register. The person would
scratch the card and reveal the discount the person would get on their purchase. The cards
offered discounts of 25%, 15%, and 10%. Gavin wanted to know the probability that
someone would receive a discount of 25% at the register. To help answer this question he
decided to keep track of all the discounts that he awarded at his register.
(a) Here’s the data Gavin collected. Complete the third column in the table.
Tally
Frequency
25% Discounts
15% Discounts
10% Discounts
(b) How many discounts did Gavin give out in total?
______
(c) What is the experimental probability that a customer would receive a 25% discount?
(d) What is the experimental probability that a customer would receive a 15% discount?
(e) What is the experimental probability that a customer would receive a 10% discount?
(f) If you can only afford to buy a television with a 25% discount, should you participate in
this Scratch and Save sale? Explain.
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7. You are going to drop a spoon 30 times from a height of 1 metre. You are going to record
the number of times the spoon lands the right-side up and the number of times it lands upside
down.
(a) Record your data in the following tally chart.
Tally
Frequency
Lands Right-side Up
Lands Upside Down
(b) How many times did you conduct the experiment?
______
(c) How many times did the spoon land right-side up?
______
(d) How many times did the spoon land upside down?
______
(e) What is the probability that the spoon will land right-side up?
(f) What is the probability that the spoon will land upside down?
8. Tylena conducted a fitness survey. She wanted to know the ages of people who exercised 3
or more times a week.
(a) She collected the following data. Complete the last column of the tally/frequency chart.
Ages
15 - 24
Tally
Frequency
25 - 34
35 - 44
45 - 54
55 - 64
65 - 74
(b) How many pieces of data does Tanya have?
_______
(c) What is the probability that people who exercise 3 or more times a week are between, and
including, the ages 15 and 24?
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(d) What is the probability that people who exercise 3 or more times a week are between, and
including, the ages 25 and 34?
(e) What is the probability that people who exercise 3 or more times a week are between, and
including, the ages 55 and 64?
9. Jerry went to the Environment Canada website and was able to collect the following data
regarding the total snowfall in Annapolis Royal during the month of February from 1987 to
2006.
Year
1987
1988
1989
1990
1991
1992
1993
Total Snowfall
(cm)
28.0
30.0
65.5
57.0
33.4
79.0
40.0
Year
1994
1995
1996
1997
1998
1999
2000
Total Snowfall
(cm)
20.6
21.5
50.0
27.0
no data
5.0
13.0
Year
2001
2002
2003
2004
2005
2006
Total Snowfall
(cm)
26.0
20.0
58.0
54.0
31.0
57.0
(a) Complete the following tally/frequency chart. Please note that the class 0 to 10 includes
0 but does not include 10.
Class
Tally
Frequency
0 - 10
10 - 20
20 - 30
30 - 40
40 - 50
50 - 60
60 - 70
70 - 80
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(b) How many pieces of data does Jerry have?
_______
(c) What is the probability that the snowfall in February will be between 50 cm and 60 cm?
(d) What is the probability that the snowfall in February will be between 20 cm and 30 cm?
(e) What is the probability that the snowfall in February will be between 0 cm and 10 cm?
(f) How likely is it that Annapolis Royal will have between 0 and 10 cm of snow in February?
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Answers:
Please note that many of the decimal and percent answers had to be rounded off.
1.
123
41
=
= 0.33 = 33%
378
126
2.
32237
= 0.25 = 25%
129098
22
11
=
= 0.12 = 12%
186
93
93
1
(b)
=
= 0.50 = 50%
186
2
3. (a)
4. (a) 4 112 052
2104661
(b)
= 0.51 = 51%
4112052
2007391
(c)
= 0.49 = 49%
4112052
71
142
=
= 0.71 = 71%
100
200
1
20
(b)
=
= 0.10 = 10%
10
200
(c) You should consider the new drug because it is likely that it could lower your blood
pressure. You know this because the probability of lowering the blood pressure is fairly
high (71%). You should still make sure that you talk to your doctor about the new drug.
5. (a)
6.
(a)
(b)
(c)
(d)
(e)
(f)
Frequency
4
11
27
25% Discounts
15% Discounts
10% Discounts
42
4
2
=
= 0.10 = 10%
42
21
11
= 0.26 = 26%
42
27
= 0.64 = 64%
42
No, there is only a very slight chance (10%) that you would be able to get the discount
you needed.
7. Answers will vary.
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8.
(a) Ages
Frequency
15 - 24
18
25 - 34
42
35 - 44
33
45 - 54
26
55 - 64
22
65 - 74
10
(b) 151
18
= 0.12 = 12%
(c)
151
42
(d)
= 0.28 = 28%
151
22
= 0.15 = 15%
(e)
151
9.
(a) Class
Frequency
0 - 10
1
10 - 20
1
20 - 30
6
30 - 40
3
40 - 50
1
50 - 60
5
60 - 70
1
70 - 80
1
(b) 19
5
(c)
= 0.26 = 26%
19
6
(d)
= 0.32 = 32%
19
1
= 0.05 = 5%
(e)
19
(f) very unlikely
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What are the Chances?
When someone gives you a probability, they are providing you with a number that describes the
chance of a particular event occurring. A probability can be written as a fraction, decimal, or
percent.
•
If there is almost no chance that the event will occur, then the probability is very close to
0. For example, the probability of selecting a king of spades from a deck of cards is
1
or approximately 0.02. If there is no chance of the event occurring, then the
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probability is equal to 0. For example, the probability that your instructor is an alien
from a distant planet is 0.
•
If there is an excellent chance that an event will occur, then the probability is very close
to 1. For example, the probability of rolling two six-sided dice and getting a sum of 3 or
35
more on those dice is
or approximately 0.97. If you are certain that a particular event
36
will occur, then the probability is equal to 1. For example, the probability that there will
be at least one birth in Nova Scotia in January is 1.
The following scale provides the descriptive terms (above the line) and their corresponding
probabilities (below the line).
Impossible
Unlikely
Half of the
Time
Likely
Certain
0
0%
0.25
25%
1
4
0.50
50%
1
2
0.75
75%
3
4
1
100%
You will use this scale to answer the following questions.
Questions:
1. Decide the chance of each one of these events occurring using the terms impossible, almost
impossible, unlikely, half of the time, likely, almost certain, certain.
(a)
(b)
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Event
You flip a coin and get a head.
Description
In an adult education classroom with twenty students at least
one student is male.
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(c)
(d)
(e)
(f)
(g)
Event
You have a bag with 7 red marbles and 2 blue marbles.
Without looking, you pull a red marble from the bag.
There will be a major snowstorm in Truro on August 12 of
this year.
When you are dealt five cards from a deck of cards, you end
up with four of a kind.
You roll a six-sided die and get a 5.
(i)
You have a safe plane trip when you travel from Halifax to
Gander.
You buy a Lotto 649 ticket and you do not win the grand
prize.
You buy a Lotto 649 ticket and you win the grand prize.
(j)
A solid iron bar will sink when it is thrown into the lake.
(k)
The next child born at the IWK will be a girl.
(l)
You roll a six-sided die and get a 3, 4, 5, or 6.
(h)
Description
(m) You roll an eight-sided die and get a 1 or 2.
(n)
Having snowfall in the first week in January in Sydney.
(o)
Having a month with 32 days.
(p)
Having a birthday in the summer.
(q)
Having a birthday in the spring or summer.
(r)
Having a birthday in any other season than winter.
2. (a) Describe a situation that is almost certain.
(b) Describe a situation that is unlikely.
(c) Describe a situation that is impossible.
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3. Draw lines to connect the event to the appropriate probability.
Event
(a) It is almost impossible to get a royal flush (A, K, Q, J,
10 from the same suit) when dealt 5 cards from a deck.
(b) Almost half of the time you should be able to pick a red
ball from a bag containing 6 red balls and 7 green balls.
(c) It is impossible for one person to have two birthdays in
the same year.
(d) It is unlikely that you would be able to pick a red ball
from a bag containing 2 red balls and 13 green balls.
(e) It is likely that you will roll a number greater than 2 on
an eight-sided die.
(f) It is certain that the sun will rise in the morning and set
at night.
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Probability
0
6
8
1
1
65000
6
13
2
15
4. You have been supplied with a deck of cards, but all the face cards (Kings, Queens, Jacks)
have been removed. Describe a situation, with these cards, that matches each description.
(a) Describe a situation that is certain.
(b) Describe a situation that is likely.
(c) Describe a situation that occurs half of the time.
(d) Describe a situation that is unlikely.
(e) Describe a situation that is impossible.
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Answers:
1. (a) half of the time
(c) likely
(e) almost impossible
(g) almost certain
(i) almost impossible
(k) half of the time
(m) unlikely
(o) impossible
(q) half of the time
(b)
(d)
(f)
(h)
(j)
(l)
(n)
(p)
(r)
almost certain
impossible
unlikely
almost certain
certain
likely
almost certain or likely
unlikely
likely
2. Answers will vary.
3. (a) matches with (iv)
(b) matches with (v)
(c) matches with (i)
(d) matches with (vi)
(e) matches with (ii)
(f) matches with (iii)
4. Answers will vary. Sample answers have been provided.
(a) certain that a card drawn from the deck is not a face card
(b) likely that a card drawn from the deck is 3 or greater
(c) half of the time a diamond or heart will be drawn from the deck
(d) unlikely that a card from the deck is a 9 or 10
(e) impossible for a face card to be drawn from the deck
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Comparing Theoretical and Experimental Probability
As you have learned, theoretical probability is based on logic and is calculated using the
following formula.
Theoretical Probability =
number of favourable outcomes
total number of possible outcomes
For example, the theoretical probability of rolling a 2 on a six-sided die is
1
or
6
0.17 or 17%.
Experimental probability is based on experiments where data is collected. This data and the
following formula are used to determine the experimental probability.
Experimental Probability =
number of times an outcome occurs
number of times the experiment is conducted
For example, suppose you rolled a six-sided die 30 times and it came up as a two 6 times. The
1
6
experimental probability of rolling a 2 would be
or or 0.20 or 20%.
5
30
1
1
Notice that the theoretical probability   of rolling a two and the experimental probability  
5
6
of rolling a two were not the same. In the questions that follow, you will be comparing
theoretical probability (what you expect to happen) and experimental probability (what actually
happens).
Questions:
Express all probabilities as a fraction, decimal, and percent.
1. (a) What is the theoretical probability of getting a head on a single flip of a fair coin?
(b) Monica flipped a coin 20 times and it came up heads 12 times. What is the experimental
probability of getting a head on a single flip of this coin?
(c) Take a coin and flip it 10 times. Record the number of times you get heads. Based on
this data figure out the experimental probability of getting a head on a single flip of this
coin?
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2. (a) What is the theoretical probability of getting a 5 on a single roll of a six-sided die?
(b) What is the theoretical probability of getting a 3 on a single roll of a six-sided die?
(c) Sonya rolled a six-sided die 60 times and obtained the following data.
Roll a 1
Roll a 2
Roll a 3
Roll a 4
Roll a 5
Roll a 6
Frequency
11
9
13
9
8
10
What is the experimental probability of getting a 5 on a single roll of this die?
(d) Using Sonya’s data, figure out the experimental probability of getting a 3 on a single roll
of this die?
(e) Take a six-sided die and roll it 18 times. Record the number of times you roll a 5. Based
on this data figure out the experimental probability of getting a 5 on a single roll of this
die?
3. Six wooden blocks of the same size and shape are placed in a bag. One block is yellow.
Two blocks are red. Three blocks are blue.
(a) What is the theoretical probability of drawing a yellow on a single draw? ____________
(b) What is the theoretical probability of drawing a red on a single draw?
____________
(c) What is the theoretical probability of drawing a blue on a single draw?
____________
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4. In the last question there was a bag with one yellow block, two red blocks, and three blue
blocks. Andrea decided to conduct an experiment using this bag of blocks. She picked a
block from the bag without looking, recorded the color, and then placed the block back in the
bag. She did this 60 times. The table below shows the data she collected.
Color of Block
Yellow
Red
Blue
Frequency
9
17
34
(a) What is the experimental probability of drawing a yellow on a single draw?
(b) What is the experimental probability of drawing a red on a single draw?
(c) What is the experimental probability of drawing a blue on a single draw?
5. Look at the theoretical and experimental probabilities that you got in the last two questions.
Were the expected values (theoretical probabilities) and actual values (experimental
probabilities) equal? Why is this so?
6. Google Search NCTM illuminations adjustable spinner. With this online simulation you
can adjust the size and number of sections on the spinner. You can repeatedly use the
spinner and examine the relationship between the theoretical probability and experimental
probability.
• Set the ‘Number of sectors’ to 3 using the -1 and +1 buttons.
• Set the ‘Number of spins’ to 50.
• Set Blue at 50%, Cyan at 25%, and Green at 25% (then press
Update).
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(a) Press ‘Spin.’ The data for those 50 spins will appear in the table. Record this
information.
Count
Experimental
Blue
Cyan
Green
Theoretical
50%
25%
25%
(b) Press ‘Spin’ a few more times until the ‘Number of spins so far’ reads 250. The data for
250 spins now appears in the table. Record this information.
Count
Experimental
Blue
Cyan
Green
Theoretical
50%
25%
25%
(c) Press ‘Spin’ several more times until the ‘Number of spins so far’ reads 1000. The data
for 1000 spins now appears in the table. Record this information.
Count
Experimental
Blue
Cyan
Green
Theoretical
50%
25%
25%
(d) For which experiment (50 spins, 250 spins, or 1000 spins) did you get the best match
between the experimental and theoretical probabilities?
(e) If you were to use the spinner 8000 times, would you expect the theoretical probabilities
and experimental probabilities to be closer or further apart? Check your answer using the
online spinner. Press ‘New experiment’, set ‘Number of spins’ to 8000, and press ‘Spin.’
Did you get what you expected?
(f) The law of large numbers states that probability statements apply in practice to a large
number of trials. In the case of this spinner simulation, one spin is considered one trial.
Look at your answers to questions (d) and (e). Do these answers support the law of large
numbers? Explain.
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7. A fair coin was repeated tossed and the following data was collected. Decimal and percent
probabilities have been rounded off in many cases.
Number of
Tosses
Expected
Number of
Heads
Actual Number
of Heads
Observed
10
5
4
100
50
59
1000
500
475
10000
5000
5093
Experimental
Probabilities
4
or 0.40 or 40%
10
59
or 0.59 or 59%
100
475
or 0.48 or 48%
1000
5093
or 0.51 or 51%
10000
Explain how the data above relates to the law of large numbers.
8. Google search maths online probability. Select What’s in Santa’s sack. In this game, you
will draw items from and then return them to Santa’s sack. You will decide how many times
this will occur. You will use the results from this simulation to predict the number of toys of
each type found in the sack.
(a) When you get the correct answer, print the page.
(b) How did the law of large numbers help you with this game?
(c) What should the theoretical probability for the drawing of each toy.
(d) We’re the experimental probabilities the same. Why or why not?
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Answers:
Many of the probabilities that were expressed as decimals and percents have been rounded off.
1
or 0.50 or 50%
2
12
3
(b)
or
or 0.60 or 60%
20
5
(c) Answers will vary.
1. (a)
2. (a)
(b)
(c)
(d)
(e)
1
or 0.17 or 17%
6
1
or 0.17 or 17%
6
8
2
or
or 0.13 or 13%
60
15
13
or 0.22 or 22%
60
Answers will vary.
1
or 0.17 or 17%
6
2
1
(b)
or
or 0.33 or 33%
6
3
3
1
(c)
or
or 0.50 or 50%
6
2
3. (a)
9
3
or
or 0.15 or 15%
60
20
17
(b)
or 0.28 or 28%
60
17
34
(c)
or
or 0.57 or 57%
30
60
4. (a)
5. The theoretical and experimental probabilities are not equal to each other but the values are
close to each other. Theoretical probability describes the likelihood something will occur. It
is not a guarantee. That is why the theoretical probability (what you expect to happen) is
close by not necessarily equal to the experimental probability (what actually happens).
6. (a), (b), and (c) - Answers will vary.
(d) Typically the 1000 spins will give the best match.
(e) The experimental should be very close to the theoretical probability when the spinner is
used 8000 times.
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(f) Notice that as we go from 50 spins to 250 spins, to 1000 spins, and finally to 8000 spins,
the experimental probabilities get closer to the theoretical probabilities. A larger number
of trials (spins) produce a better match between the experimental and theoretical
probabilities. This is exactly what the law of large numbers states.
7. As the number of tosses (trials) increases, the experimental probability gets closer to the
theoretical probability of 0.50 or 50%. One can say that the theoretical statement of 0.50 or
50% applies to a large number of trials (tosses). This is the law of large numbers.
8. (b) If you don’t have enough trials then your experimental probabilities will not be very close
to the theoretical probabilities and it is unlikely that you will be able to figure out how
many of each toy are hidden in the sack. You need a large number of trials. That means
that you are using the law of large numbers to solve this problem.
(c) Answers will vary based on which game you selected.
(d) Probably not
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Real World Probability
Over the last few lessons you’ve looked at all sorts of theoretical and experimental probabilities.
Many of the questions dealt with flipping coins, drawing cards, or spinning spinners. Although
these types of questions are useful when trying to understand probability, these contexts don’t
represent.
1. Anne and Dave had one healthy child. They were thinking of having a second child when
Anne learned that she had a connective tissue disorder. Her doctor told her that, due to this
disorder, there was a 25% probability that her child would be born with significant heart
defects. How concerned should Anne and Dave be about this new information? In your
opinion, should they try to have a second child? Explain your reasoning.
2. Colin spent much of his youth doing outdoor sports where he had excessive exposure to the
sun. At the age of 35, his doctor informed him that a small growth on his forehead was a
form of skin cancer. The doctor told him that there was a 99% survival rate for this form of
cancer. How concerned should Colin be about his chances for survival if he gets treatment
for the cancer? Explain your reasoning.
3. Statistics Canada reported that in 2007 there were 17 homicides and 3866 assaults for every 1
million people in Canada.
(a) What was the probability of being the victim of a homicide in 2007?
(b) What was the probability of a being the victim of an assault in 2007?
(c) Are you concerned with the probabilities you calculated in questions (a) and (b). Would
you change your way of living based on these probabilities?
(d) Do you think that these probabilities can be applied to anyone living anywhere in
Canada? Explain.
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4. The probability of sustaining a head injury in a motorcycle accident when a helmet is not
worn is 0.49. The probability of sustaining a head injury with a helmet is 0.29. Do you
agree with the law that all riders must wear helmets? Explain.
5. Go to YouTube. Search for the video titled probability 0.0000001 %. View the video.
What is the significance of using the number 0.0000001% in the title of the video?
6. In James Bond movies, James regularly performs incredible tasks. In many of his movies, he
plays poker against the villain and ultimately wins. In one movie, his winning hand is a royal
flush (Ace, King Queen, Jack and 10 of the same suit). The probability of being dealt these 5
cards is 1 in 65 000. If you were James, would you rest your own fate and that of your
country on this kind of probability? Explain.
7. Did you know that in a group of 23 people, the probability of at least two people sharing the
same birthday is slightly greater that 0.5. Are you surprised by this? What would happen if
there were 100 people in the group? Explain your reasoning.
8. The chances of selecting the winning numbers with the purchase of one ticket in Lotto 649 is
1in 13 983 816. Suppose you could buy 13 983 816 tickets, all with different numbers? You
would be guaranteed to win but would this be a wise decision? Why or why not?
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9. Do your best to match each of these situations to the appropriate probability. Some of these
might surprise you.
(a)
(b)
(c)
(d)
(e)
(f)
Killed in Commercial Plane Crash
Die from AIDS
Killed by a Hurricane
Die from Cancer
Die in a Car Accident
Killed by Lightning
___
___
___
___
___
___
1 in 500
1 in 6000
1 in 11 000
1 in 1.4 million
1 in 6 million
1 in 10 million
10. A major coffee shop chain has a promotion where there is a 1 in 10 chance of winning a free
muffin with the purchase of a coffee. If the company sells 60 000 cups of coffee during the
promotion, how many free muffins should they expect to give away?
11. Create your own real-world situation that might result in a probability of 0.25.
12. Angela knows that the probability of winning the grand prize in Lotto 649 is 1 in 13 983 816.
She wants to try to try to visualize this. She is going to use Post-It notes to do this. She
knows that a small stack of Post-It notes has 100 sheets and measures 1.1 cm in thickness.
(a) She wants a stack of Post-It notes that has 13 983 816 sheets in it. How thick (tall) will
this stack be? Select the correct answer.
(i) 1.54 km tall
(ii) 0.53 km tall
(iii) 103.7 m tall
(iv) 8.92 m tall
(b) Her friend will place an X on one of the 13 983 816 sheets to represent the grand prize.
What is the likelihood that Angela will be able to randomly select that sheet with the X
on it from stack of 13 983 816 sheets?
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Answers:
1. There is a 1 in 4 chance that the child will be born with a heart defect. That means that there
is a 3 in 4 chance that the child will not have a heart defect. Some people would argue that
since Anne and Dave already have one healthy child, that the risk is too high in having a
second child. Others would state that the risk is low enough to justify trying to have a second
child. There isn’t a clear answer to this question.
2. A 99% survival rate is very good. It means that there is only a 1% chance that Colin will not
survive the treatment for skin cancer. There is only a very slight possibility that he wouldn’t
survive the treatment. He shouldn’t be very concerned but he should still seek immediate
treatment.
17
1 000 000
3866
(b) Probability =
1 000 000
(c) No, these probabilities are very low.
(d) No, people living in more disadvantaged areas or engaging in illegal activities might be
subject to higher probabilities of assault and/or homicide.
3. (a) Probability =
4. The probability of sustaining a head injury when no helmet is worn is 49% or almost 1 in 2.
That is an alarming probability. That probability reduced significantly when a helmet is
worn. When you consider the burden to the family of the accident victim, the victim’s
quality of life, and the cost to health care system, it seems that logical that helmets should be
worn. Although helmets do not ensure complete safety, they do reduce the risk significantly.
5. It is virtually impossible for this sequence of events to occur that is why it is titled
“probability 0.0000001 %.” It seems to be an appropriate title. You can probably say that is
sequence of events would never occur. If so, maybe it should be titled “probability 0%.”
6. The probability of 1 in 65 000 is very close to 0%. That means that it is almost impossible to
be dealt a royal flush. It hardly seems worsh it for James to risk his own safety and the
security of his country on such a slim chance.
7. As the number of people increases, the probability of having two people with the same
birthday increases.
8. You would win but there is still a possibility that you would have to share your winning with
someone else who selected the same numbers. You also have the problem that you would
initially need $13 983 816 to buy the 13 983 816 tickets. If you had over $13 million dollars,
would you want to risk it in this manner? There is also the problem of buying over 13
million tickets. Even if it took only 5 seconds to purchase a single ticket, it would take over
809 days (2.2 years) to purchase over 13 million tickets. Lotto 649 draws occur every week.
You don’t have enough time to buy the 13 million tickets.
NSSAL
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9.
(d)
(e)
(b)
(f)
(c)
(a)
1 in 500
1 in 6000
1 in 11 000
1 in 1.4 million
1 in 6 million
1 in 10 million
10. 6000 muffins
12. (a) (i) 1.54 km tall
(b) almost impossible
NSSAL
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C. D. Pilmer
The Lotto 649 Activity
Have you ever played Lotto 649?
Yes ___
No ___
If you answered yes, do you play it regularly?
Yes ___
No ___
If you play it regularly, how much do you spend per month on Lotto 649?
Less than $5 ___
Between $5 and $15 ___
More than $15 ___
You are now going to select 6 numbers between 1 and 49. Do not select the same number twice.
Put these 6 numbers from small to largest in the first six boxes below. From the remaining 43
numbers that you did not choose, choose one more. This will be your bonus number. Fill it in
the last box.
Bonus
Number
These numbers will be your Lotto 649 numbers. Your instructor will now supply you with
sheets that have all the winning numbers for Lotto 649 from June 1982 until June 2008
(Obtained from http://www.lotterybuddy.com/l6list82.htm ). That is 26 years of Lotto 649
winning numbers from 2528 drawings. Your mission is to compare your numbers to the winning
numbers on the sheets and see if you would have won anything. Record the information in the
table below.
Matches
Number of Times the
Matches Occurred
All 6 numbers match
Dates that the
Matches Occurred
Prize
(Fictitious)
$16 000 000
5 out of 6 plus the
bonus number
$120 000
5 out of 6 matches
$2 500
4 out of 6 matches
$75
3 out of 6 matches
$10
2 out of 6 plus the
bonus number
$5
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You bought 2528 tickets over the 26 years. If each ticket costs $2, how much would you have
spent in total?
How much money did you win?
For Your Information
Here are the probabilities of winning each of the prizes in Lotto 649.
1
Probability of Matching 6 of 6 (Grand Prize) =
13 983 816
6
1
Probability of Matching 5 of 6 plus Bonus # =
or
13 983 816
2 330 636
252
1
Probability of Matching 5 of 6 =
or approximately
13 983 816
55 491
13 545
1
Probability of Matching 4 of 6 =
or approximately
13 983 816
1032
246 820
1
Probability of Matching 3 of 6 =
or approximately
13 983 816
57
172 200
1
Probability of Matching 2 of 6 plus Bonus # =
or approximately
13 983 816
81
Total Number of Potential Prizes = 1 + 6 + 252 + 13545 + 246820 + 172200
= 432 824
Total Number of Different Tickets = 13 983 816
Probability of Winning a Prize =
432 824
or approximately 0.03 or 3%
13 983 816
That means that you lose approximately 97% of the time when you play Lotto
649.
Watch the following YouTube video.
http://www.youtube.com/watch?v=WSZxADY4yE8&list=PL36625D82F83D970F&index=4
&feature=plpp_video
(or Google Search: YouTube Understanding the Chances of Winning Lotto 6 49 Pilmer)
NSSAL
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C. D. Pilmer