Simulation

What if you can’t perform an experiment to find the probability of a certain event?

▪Say there’s a young family who plans to have 3 children. The mom wants to have all boys, the dad wants to have all girls. What is the probability that all 3 of their children will have the same gender?

▪ If it is just as likely that a boy is born that a girl is born, what is a way we could conduct an experiment to find the probability without having to track birth rates?

Simulation

▪Simulation -procedure that will allow you to answer questions about real problems by running experiments that closely resemble the real situation.

▪Results from the simulations are used to estimate probabilities of problems we otherwise could not easily solve.

▪Trial- each round of a simulation

▪Success- each trial that had the desired outcome

▪Failure- each trial that did NOT have the desired outcome

Simulation Vocabulary

Five Steps in Simulation

1. Define the basic outcome of the real experiment

2. Chose a way to simulate the experiment and define what each outcomes means in the real experiment.

3. Define the trial and what that would be in the real experiment

4. Define what is meant by a success in each trial

5. Perform n trials (n will be whatever number is needed for that specific example)

Example

▪We can use simulation to find the probability that the young family will have either all three boys or all three girls.

1. Basic Outcome: Gender of children born

2. Simulation: perform 3 coin tosses many times to indicate lots of different possible combinations for all 3 children. A heads (H) means a boy is born and a tails (T) means a girl is born.

3. Trial: each sequence of three tosses will represent all 3 children being born

4. Success: If a trial results in either HHH or TTT, it represents all boys and all girls, which is what we want so we call that a success.

▪ If a trial results in any other order of H’s and T’s, then it is called a failure. (ex: HTH or TTH)

The estimate for the probability

5. Suppose trials are performed, resulted in either HHH or TTT.

The estimated probability .

According to our estimation there is a 28% chance that the family has either three boys or three girls.

Example

More on SimulationWays to Simulate Experiments

▪ How could a number cube or a deck of cards be used to simulate having a boy or a girl?

Number Cube: odds could represent boys, even could be girls; you could roll the cube 3 times for each trial

Cards: black cards might be girls, red could be boys; you could draw 3 cards for each trial

▪What if you needed to simulate a 10% chance that a car tire might pop?

Colored chips in a bag, could be a certain color that represent a popped tire and you could draw 4 chips for each trial (one for each tire on a car)

Setup a SimulationCreate a simulation to model the next 3 flights being

overbooked if 20% of the flights are overbooked.1. Basic Outcome:

2. Simulation:

3. Trial:

4. Success:

Flight Status

Draw beads from a bag. 20% of the beads will be pink.

Draw 3 beads.

All 3 beads are pink.

Practice