MDM4U
4.1 Simulation and Experimental Probability
Date __________
Conditions for a “fair game”
 a game is fair if…
o all players have an equal chance of winning
o each player can expect to win or lose the same number of times in the long run
o each player's expected payoff is zero
Important vocabulary
Trial: one repetition of an experiment
e.g., flip a coin; roll a die; flip a coin and spin a spinner
Random variable: a variable whose value corresponds to the outcome of a random event
Expected value: the value to which the average of a random variable’s values tends after many
repetitions; also called the average value or mean value
Event: a set of possible outcomes of an experiment
Simulation: an experiment that models an actual event
Experimental Probability is the observed probability (also known as the relative frequency)
of an event, A, in an experiment. It is found using the following formula:
Note: probability is a number between 0 and 1 inclusive.
It can be written as a fraction or decimal.
MDM4U
4.1 Simulation and Experimental Probability
Date __________
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 1/13, 1/52
Spin a spinner-any (realistically 12 or fewer)
Ex.: Describe a simulation that models:
a) A hockey player who scores on 17% of the shots he takes
b) A baseball player’s batting average is 0.300
c) A randomly chosen student has a birthday during the school year
Solution:
a) Roll a die. Let 1 represent a goal.
b) Put 3 red balls and 7 blue balls in a garbage can. Drawing a red ball represents a hit.
c) Roll a die. Any number other than 1 represents the student having a birthday during the school
year.
Ex.: If we wanted to determine the probability of a newlywed couple (who plans on having 3
children) having two boys and then a girl it would be impractical to go out and find 100 couples and
have them notify you of the gender of their child each time they have one...
Instead we could use a coin to simulate having a child. Maybe "heads" will represent having a girl and
"tails" will represent having a boy. To conduct our experiment a trial would consist of tossing a coin
three times and recording the "gender" of each child. After conducting many times we could
determine the probability of getting tails tails heads, which would represent having two boys and
then a girl.
Flipping coins can be time-consuming, so we can use various technology to help us.
We will use the Graphing Calculator TI-83.
MDM4U
4.1 Simulation and Experimental Probability
Date __________
This function generates random integers and requires three inputs:
randInt(smallest integer possible, largest integer possible, how many integers to generate)
For example, randInt(1,5,10) would generate 10 random integers between 1 and 5.
Each time you hit enter on the calculator you will get a new set of 10 integers.
Ex.: During March break there is typically a 20% chance of rain each day. What is
the probability that it will rain on less than three of the 7 days?
Since 20% can be expressed as 1/5 as a fraction in lowest terms we can have the calculator generate
numbers between 1 and 5. If a 5 comes up that will represent a rainy day. The calculator should
generate 7 numbers to represent each day of the week. After many trials we can determine how
many sets of 7 numbers have the number 5 show up less than three times.
MDM4U
4.1 Simulation and Experimental Probability
Date __________
4 out of these 6 trials had less than 3 "rainy days".
You should do many more trials but at this point you would say that there is a 67% chance that less
than three of the seven days of the March break will have rain.