Welcome to MDM4U (Mathematics of Data
Management, University Preparation)
http://www.wordle.net/
Minds on

What does it mean when you give someone a
choice between 2 options and they respond
with “flip a coin”?
Overall Expectations - Probability


1. Solve problems involving the probability of
an event, or a combination of events, when
there is a fixed number of outcomes
2. Solve problems involving the counting of
ordered and unordered objects to determine
the probability of an event
Introduction to Simulations
and Experimental Probability
Learning goals:
Design a simulation for a real-world event
Calculate experimental probability
MSIP / Home Learning:
Read through Example 2 - solution 1, p. 207
Complete pp. 209-212 #1, 5, 8-10, 12-13
Investigation – Experimental Probability



How does flipping a coin relate to the gender
of a baby?
How likely is it that a family with 3 children
has 3 boys?
How could you use coins to simulate this?
Baby Simulation





Decide which gender each outcome (heads and
tails) will correspond to.
Flip 3 coins. Record the number of heads.
 This represents one family with 3 children
Repeat 19 more times.
Calculate the experimental probability of 0, 1, 2,
3 heads.
Record your results in a table.
Number of Heads, X
Tally
Frequency
P(X)
0
II
2
2 / 20
1
IIIII II
7
7 / 20
2
IIIII III
8
8 / 20
3
III
3
3 / 20
Our results…
Team
1
2
3
4
5
6
7
8
9
10
11
12
Total
0
4
2
3
2
1
3
1
2
2
4
5
3
32
1
5
7
9
11
11
10
6
11
10
7
7
7
102
2
9
8
6
7
2
5
11
4
4
10
7
6
82
3
2
3
2
0
4
2
3
3
4
2
1
4
33
Reflect: How many different outcomes are there?
Do they have different probabilities?
Conditions for
a “fair game”

a game is fair if…



all players have an equal
chance of winning and
equal payouts, or
each player can expect to
win or lose the same
number of times in the
long run with equal
payouts, or
each player's expected
payoff is zero
http://www.math.psu.edu/dlittle/java/probability/plinko/index.html
http://probability.ca/jeff/java/utday/
A standard deck of cards
Are the following games fair?


Mike and Ike…
Roll a die.



Roll a die.



If an odd number shows, Mike pays Ike $1.
If an even number shows, Ike pays Mike $2.
Draw cards with replacement.



If a 1, 2 or 3 shows, Mike pays Ike $1.
If a 4, 5 or 6 shows, Ike pays Mike $1.
If it’s a red card, Mike pays Ike $1.
If it’s a spade, Ike pays Mike $1.
Flip 3 coins.


If 3 tails show, Mike pays Ike $7.
Otherwise, Ike pays Mike $1.
Important vocabulary

Trial: one repetition of an experiment

Random variable: a variable whose value
corresponds to the outcome of a random
event
More Vocabulary



Expected value: the value to which the
average of a random variable’s values tends
after many repetitions; also called the
average value
Event: a set of possible outcomes of an
experiment (e.g. drawing a heart)
Simulation: an experiment that models an
actual event (e.g. flip a coin to simulate the
gender of a baby)
Probability



A measure of the likelihood of an event
Based on how often a particular event occurs
in comparison with the total number of trials
Probabilities derived from experiments are
known as experimental probabilities
Experimental Probability



The observed probability of an event, A, in an
experiment.
Denoted P(A)
Found using the following formula:
P(A) = number of times A occurs
total number of trials
Note: probability is a number between 0 and 1 inclusive.
It can be written as a fraction, decimal or percent.
Simulations




A simulation is an experiment that has the
same probability as an actual event.
Flip a fair coin½
Roll a fair die  1/6, 2/6, 3/6, 4/6, 5/6
Draw a card from a standard deck (52)




½, ¼, x/52, x/13
Many others if you use a partial deck
Hold a draw  any
Spin a spinner  any (realistically 12 or fewer)
Example 1







Describe a simulation that models:
a) A hockey player who scores on 1/6 or 17% of the shots he
takes takes 6 shots in a game
b) A baseball player whose batting average is 0.300 gets 4 atbats in a game
c) a student in the class has a birthday during the school year
a) Roll a die. Let 1 represent a goal. Roll the dice 6 times.
b) Put 3 red balls and 7 blue balls in a garbage can. Drawing
a red ball represents a hit. Draw 4 balls with replacement.
c) Roll a die. Any number other than 1 represents the student
having a birthday during the school year.
MS Excel Formulas to generate random integers

Random 0 or 1 (coin toss, predict gender of a baby)
=ROUND(RAND()*(1),0)

Random H or T (0 or 1)
=IF(ROUND(RAND()*(1),0)=0,"H","T")

Random integer between 1 and 5 (football kicker p. 211 #10)
=ROUND(RAND()*(4)+1,0)

Random integer between 1 and n
=ROUND(RAND()*(n-1)+1,0)


Type a formula into a cell, then copy and paste to a group of cells to
simulate multiple trials e.g., 4.1 random numbers.xls
Press F9 instead of ENTER to generate a static random number
Recap
MSIP / Home Learning


Read through Example 2 - solution 1, p. 207
Complete pp. 209-212 #1, 5, 8, 9-10, 12-13
Warm up

Toronto Blue Jays shortstop Jose Reyes has
a lifetime 0.286 batting average. On average
he gets a hit in 2 out of every 7 at bats.
Design a simulation to determine whether he
gets a hit in the Blue Jays’ next game (5 at
bats).
Solution
1.
2.
To simulate one at-bat, we need an experiment where an event
has a 286/1000 (or 2/7) probability. This could be any ONE of the
following:
 Put 1 000 numbered balls in a drum. Choose a ball. Balls from 1
to 286 represent a hit. Replace the ball.
 Generate a random number from 1 to 1 000 (or 1 to 7). Any
number from 1 to 286 (or 1-2) represents a hit.
 Roll a 7-sided die – 1 or 2 is a hit
 Spin a spinner divided into 7 segments of 360°÷7 = 51.43°.
Colour two green – those sections represent a hit.
 Remove all cards 8 or higher from a deck (aces low). Draw a card
from the cards that remain. An A or 2 represents a hit. Replace
the card.
Repeat 5 times to simulate 5 more at-bats.
Theoretical Probability
Learning goal: Calculate theoretical probabilities
Questions? pp. 209-212 #1, 5, 8-10, 12-13
MSIP / Home Learning: pp. 218-219 # 4-7, 9, 10, 12
Have you seen this store?
Gerolamo Cardano





Born: 1501, Pavia, Italy
Died: 1571 in Rome (on
the date he predicted
astrologically)
Physician, inventor,
mathematician, chess
player, gambler
Invented combination
lock, Cardan shaft
Published solutions to
cubic and quartic
equations
Games of Chance

Most historians agree that the modern study of
probability began with Gerolamo Cardano’s
analysis of “Games of Chance” in the 1500s.

http://en.wikipedia.org/wiki/Gerolamo_Cardano
http://www-gap.dcs.st-and.ac.uk/~history
/Mathematicians/Cardan.html

A few terms…

simple event: an event that consists of
exactly one outcome (e.g., rolling a 3)

sample space: the collection of all possible
outcomes of an experiment (e.g.,
{1,2,3,4,5,6} for rolling a die)

event space: the collection of all outcomes of
an experiment that correspond to a particular
event (e.g. {2,4,6} are the even rolls of a die)
General Definition of Probability

assuming that all outcomes are equally likely,
the probability of event A is:
P(A) = n(A)
n(S)

where n(A) is the number of elements in the
event space and n(S) is the number of
elements in the sample space.
Example #1

When rolling a single die, what is the
probability of…

a) rolling a 2?
b) rolling an even number?
c) rolling a number less than 5?
d) rolling a number greater than or equal to 5?



Example #1a

When rolling a single die, what is the
probability of…

a) rolling a 2?
A = {2},
S = {1,2,3,4,5,6}
P(A) = n(A) = 1 = 0.17
n(S)
6
Example #1b

When rolling a single die, what is the
probability of…

b) rolling an even number?
A = {2,4,6},
S = {1,2,3,4,5,6}
P(A) = n(A) = 3 = 1 = 0.5
n(S)
6
2
Example #1c

When rolling a single die, what is the
probability of…

c) rolling a number less than 5?
A = {1,2,3,4},
S = {1,2,3,4,5,6}
P(A) = n(A) = 4 = 2 = 0.67
n(S)
6
3
Example #1d

When rolling a single die, what is the
probability of…

d) rolling a number greater than or equal to 5?
A = {5,6}, S = {1,2,3,4,5,6}
P(A) = n(A) = 2 = 1 = 0.33
n(S)
6
3
The Complement of a Set





All outcomes in the sample space that are not in
the set A.
Written A’
Read A-complement or A-prime
n(A) + n(A’) = n(S)
P(A) + P(A’) = 1

So, P(A’) = 1 – P(A)
A standard deck of cards
Example #2a

When selecting a single card from a standard
deck (no Jokers), what is the probability you
will pick…





a) the 7 of Diamonds?
b) a Queen?
c) a face card (J, Q or K)?
d) a card that is not a face card?
e) A red card = 26 / 52 = 1 / 2
Example #2a

When selecting a single card from a standard
deck (no Jokers), what is the probability you
will pick…

a) the 7 of Diamonds?
P(A) = n(A) = 1 = 0.02
n(S)
52
Example #2b

When selecting a single card from a standard
deck, what is the probability you will pick…

b) a Queen?
P(A) = n(A) = 4 = 1 = 0.08
n(S)
52 13
Example #2c

When selecting a single card from a standard
deck, what is the probability you will pick…

c) a face card (J, Q or K)?
P(A) = n(A) = 12 = 3 = 0.23
n(S)
52 13
Example #2d

When selecting a single card from a standard
deck, what is the probability you will pick…

d) a card that is not a face card?
P(A) = n(A) = 40 = 10 = 0.77
n(S)
52 13
Example #2d (cont’d)

Another way of looking at P(not a face card)…


we know: P(face card) = 3
13
and, we know: P(A’) = 1 - P(A)

So…
P(not a face card) = 1 - P(face card)
P(not a face card) = 1 - 3 = 10
13 13
Example #2e

When selecting a single card from a standard
deck, what is the probability you will pick…

e) a red card?
P(A) = n(A) = 26 = 1
n(S)
52
2
MSIP / Home Learning

pp. 218-219 # 4-7, 9, 10, 12
Next class: A look at Venn Diagrams