10.2 Theoretical Probability
Try These
You will need
i) (0.5)(3) 2 (0.5)(2) 5
• plain paper
ii) (0.6)(5) 1 (0.2)(23) 5
2.4
In Lesson 10.1, you rolled dice to represent a car recall. How
does the theoretical probability of rolling 1 compare with your
experimental probability?
1
Record the possible rolls for two dice in the outcome chart.
Circle rolls with at least one 1. What is the theoretical
probability of rolling at least one 1?
Second die
First die
theoretical
probability
the ratio of the
number of
favourable
outcomes to the
total number
of equally
likely possible
outcomes
0.5
1
2
3
4
5
6
1
1, 1
1, 2
1, 3
1, 4
1, 5
1, 6
2
2, 1
2, 2
2, 3
2, 4
2, 5
2, 6
3
3, 1
3, 2
3, 3
3, 4
3, 5
3, 6
4
4, 1
4, 2
4, 3
4, 4
4, 5
4, 6
5
5, 1
5, 2
5, 3
5, 4
5, 5
5, 6
6
6, 1
6, 2
6, 3
6, 4
6, 5
6, 6
P(rolling 1) 5
favourable outcomes
possible outcomes
rolls with 1
all possible rolls
5
11
, or 30.555… %
36
The theoretical probability of rolling at least one 1 is
Reflecting
What happens to
the experimental
probability as the
number of trials
increases?
252
2
31 %.
How does the theoretical probability compare with the
experimental probability for your class?
e.g., The experimental probability for the class was close to the
theoretical probability.
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5
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Example 1
A store has a scratch-and-win promotion. Each card has two
columns. Each column has one hidden star. Scratch one box per
column. If you uncover both stars, you win a prize. Suppose that you
scratched eight cards. How many prizes could you expect to win?
Solution
A. Complete the tree diagram. What is the
theoretical probability of winning a prize?
There are 4 possible outcomes.
The theoretical probability of winning
1
, or 25 %.
a prize is
4
Column 1
1
b(8), or
4
You could expect to win
2
2
Outcomes
star
star, star - win
no star
star, no star - lose
star
no star, star - lose
no star
no star, no star - lose
star
no star
Reflecting
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Expected prizes 5 (probability of winning)(number of cards)
5a
1
Column 2
B. How many prizes could you expect to win if you scratched
eight cards?
Scratch and Win!
2
prizes if you scratched 8 cards.
Suppose that you
scratched four
cards. Could you
be certain to win a
prize? Explain.
Example 2
Lee tossed a coin 4 times. She got tails each time. Who is correct?
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• Lee says she will get tails on toss 5 because the coin keeps
landing this way.
• Tim says the coin is more likely to land on heads because it is
unlikely to land on tails 5 times in a row.
AW12
• Amit says there is an equal chance
of the coin landing on
0176519637
heads or tails on toss 5.
Figure Number
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Company
MPS
Solution
Technical
Pass
1st pass
A. What is the experimental probability
of tossing
tails 4 times in
Approved
a row?
Not Approved
4
favourable outcomes
AW125
, or
number of trials
01765196374
Figure Number
100 %
C10-F17-AW12.ai
Company
B. What is the theoretical
probabilityMPS
of tails on each toss?
Technical
1
favourable outcomes
Pass
5
, or
possible outcomes
Approved 2
1st pass
50
Not Approved
C. Who is correct about toss 5?
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10_AW12_Ch10.indd 253
Amit
%
is correct.
Reflecting
What do you think
would happen to
the experimental
probability if Lee
did 100 trials?
Chapter 10 Probability
253
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Practice
1. Which event is more likely: tossing tails with a coin or rolling 3
with a standard die? Why?
P(tossing tails) 5
1
1
, or 50% P(rolling 3) 5 , or 16.666…%
2
6
Tossing tails is more likely since 50% is greater than 16.666…%
1
2
3
4
2. In Lesson 10.1 you played Slahal.
a) Suppose that the team guessing said “hand 1 and
hand 3.” What other guesses are possible?
hands 1 and 4, hands 2 and 3, and hands 2 and 4
b) What is the theoretical probability of guessing correctly?
P(guessing correctly) 5
5
favourable outcomes
possible outcomes
1
, or 0.
4
25
, or
25
%
c) How does the theoretical probability of guessing correctly
compare with the experimental probability from your game
in Lesson 10.1?
e.g., We guessed correctly 20% of the time. This is close to the theoretical probability of 25%.
d) Suppose you played more rounds of the game. What
would happen to the experimental probability?
e.g., The experimental probability of guessing correctly would likely get closer to 25%.
3. Rousell runs a duck pond game at a fair. The pond has
75 ducks. Each duck is marked with the size of prize.
For each turn, the player takes 1 duck.
There are 60 ducks marked with “small,” 12 ducks marked
with “medium,” and 3 ducks marked with “large.”
a) What is the theoretical probability of winning each prize?
Write your answer as a fraction, a decimal, and a percent.
60
P(small) 5
, or 0.8, or 80%
75
12
P(medium) 5
, or 0.16, or 16%
75
3
P(large) 5
, or 0.04, or 4%
75
254
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b) Abby says that she could use a tree diagram to determine
the probability of winning two large prizes if you played
twice. Do you agree? Explain.
No. e.g., The probabilities of winning are different for each
size of prize. You can only use a tree diagram if there is
the same chance of winning each size of prize.
4. Jaaji used this spinner for an experiment. For each trial,
he spun twice and recorded the sum of the two numbers.
He did 50 trials.
3
b) What is the theoretical probability of the sum being a
multiple of 3?
P(multiple of 3) 5
Hint
Multiples of 3 are
3, 6, 9, 12, 15,…
favourable outcomes
total outcomes
4. a) e.g.,
First spin
4
5 , or 44.444…%
9
Second spin
6
8
a) What are all the possible outcomes for one trial? Use a
sheet of paper. Make a chart or a tree diagram.
The theoretical probability is about 44%.
c) About how many times is it likely that Jaaji spun a multiple
of 3? Why?
3
3
6
8
6
9
11
6
9
12
14
8
11
14
16
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e.g., (0.444…)(50 trials) 5 22 He likely spun a multiple of 3 about 22 times.
5. Toby put a new roof on his house. He knows that when it
rains, there are two possible outcomes. The roof will leak, or
the roof will not leak.
1
The probability that the roof will leak is not 2. Why not?
e.g., The probability is not
1
2
because the two outcomes are
not equally likely. If the roof was done properly, it should not
leak for a long time.
AW12
1
6. The theoretical probability of 0176519637
winning a draw is 10. Suppose
Figure Number
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that you buy 10 tickets for the draw.
Company
Are you certain to win a prize?
Explain.
Technical
MPS
Pass
1st pass
No. e.g., The theoretical probability
means that
10 of the
1
tickets sold will win a prize.
9Approved
the tickets
Notof
Approved
10
will not win.
All your tickets could be tickets that will not win.
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Chapter 10 Probability
255
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