Math Core

Some probabilities are too hard to count and too slow or costly to test for real. What is the chance a basketball player makes at least $2$ of her next $3$ shots? You can't make her shoot thousands of times, but you can act it out with a coin, a number cube or random digits. That's a simulation.

What is a simulation?

Definition

Simulation

A simulation is an experiment that uses a simple random tool to model a real situation. The tool's outcomes must have the same probabilities as the real outcomes. You run many trials, then use the experimental probability as an estimate.

Choosing a random tool

Match the tool to the probability you need.

Probability of the real event	A tool that works
$\dfrac{1}{2}$	flip a coin: heads means the event happens
$\dfrac{1}{6}$ or $\dfrac{1}{3}$	roll a number cube: a $1$ means yes (or $1$ or $2$ for $\dfrac{1}{3}$ )
$\dfrac{1}{4}$	spin a spinner with $4$ equal sections, one of them shaded
a percent like $30\%$ or $60\%$	pick random digits $0$ to $9$ : let $3$ digits (or $6$ digits) mean yes

Random digits are the most flexible tool. Each digit from $0$ to $9$ has a $\dfrac{1}{10}$ chance, so you can match any probability in tenths. Computers and calculators can produce long strings of random digits for you.

Running a simulation

Four steps of a simulation

Choose a tool whose outcomes have the right probabilities.
Define a trial: what one run of the situation looks like with your tool.
Run many trials and record whether the event happened in each.
Estimate: the probability is about $\dfrac{\text{trials where the event happened}}{\text{total trials}}$ .

Worked example: Three shots

Keisha makes $60\%$ of her shots. Use a simulation to estimate the probability that she makes at least $2$ of her next $3$ shots.

Tool: random digits. Let $0, 1, 2, 3, 4, 5$ mean "make" (that's $6$ of $10$ digits, or $60\%$ ) and $6, 7, 8, 9$ mean "miss."
Trial: one group of $3$ digits stands for $3$ shots.
Run trials. Here are $10$ trials.

Trial	Digits	Makes	At least $2$ ?
$1$	$381$	$2$	yes
$2$	$774$	$1$	no
$3$	$059$	$2$	yes
$4$	$628$	$1$	no
$5$	$441$	$3$	yes
$6$	$963$	$1$	no
$7$	$207$	$2$	yes
$8$	$855$	$2$	yes
$9$	$196$	$1$	no
$10$	$530$	$3$	yes

Estimate. The event happened in $6$ of the $10$ trials, so the probability is about $\dfrac{6}{10} = 0.6$ .

Ten trials give only a rough estimate. Just like any experiment, running more trials (a class might combine results to get $200$ ) gives an estimate you can trust more.

Common mistake

The tool must match the real probabilities, not just the number of outcomes. A shot is either a make or a miss, but you can't simulate Keisha's shots with a coin, because a coin gives a make only $50\%$ of the time, not $60\%$ .

Worked example: Guessing on a quiz

A quiz has $4$ true-or-false questions. Explain how to simulate the chance that someone who guesses on every question gets all $4$ right.

Each guess is right with probability $\dfrac{1}{2}$ , so flip a coin: heads means a correct answer. One trial is $4$ flips. Run many trials and count the ones that are all heads. Divide that count by the number of trials.

(You can check the result with the tree-diagram method: $16$ outcomes, one of them all heads, so the theoretical probability is $\dfrac{1}{16}$ . A good simulation should give something close to that.)

Tip

Before you start, write down what each outcome of your tool means, like " $0$ to $5$ = make." Deciding in the middle of the simulation makes mistakes easy.

Practice

Practice 1

A basketball team wins half of its games. Which tool is best for simulating whether the team wins its next game?

Practice 2

A bus is on time $70\%$ of the time. You want to simulate one day using a random digit from $0$ to $9$ . How many of the ten digits should stand for "on time"?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

A class ran a simulation $50$ times. The event they were studying happened in $18$ of the trials. What is their estimate of the probability? Give your answer as a decimal.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

A carnival game is won $\dfrac{1}{3}$ of the time. Which simulation does not model the game correctly?

Practice 5

To simulate families with two children, Nia flips a coin twice for each family (heads = girl, tails = boy). Her $20$ trials are below.

GB, BB, GG, BG, GB, GG, BB, BG, GB, BB, GG, GB, BG, BB, GB, GG, BG, BB, GB, BG

Based on her simulation, what is the probability that a family with two children has two girls? Give your answer as a fraction.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 6

A softball player gets a hit $30\%$ of the time. To simulate $3$ at-bats, you use groups of $3$ random digits, where $0$ , $1$ and $2$ mean a hit and $3$ through $9$ mean no hit. Here are $12$ trials.

$482 \quad 937 \quad 615 \quad 789 \quad 206 \quad 358 \quad 974 \quad 841 \quad 663 \quad 095 \quad 587 \quad 719$

Based on these trials, what is the probability that she gets at least one hit in $3$ at-bats?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

A simulation estimates that the probability of a traffic jam on Main Street during a school morning is $0.25$ . About how many of the $180$ school mornings in a year should have a traffic jam?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.