### Beyond AP Statistics

#### Probability Basics

#### Small Samples

#### Distributions

#### Power

### Beyond AP Statistics

#### Probability Basics

#### Small Samples

#### Distributions

#### Power

# Simulation of Random Events

**Simulation** is a way to model random events, such that
simulated outcomes closely match real-world outcomes. By
observing simulated outcomes, researchers gain insight on
the real world.

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## Why use simulation?

Some situations do not lend themselves to precise mathematical treatment. Others may be difficult, time-consuming, or expensive to analyze. In these situations, simulation may approximate real-world results; yet, require less time, effort, and/or money than other approaches.

## How to Conduct a Simulation

A simulation is useful only if it closely mirrors real-world outcomes. The steps required to produce a useful simulation are presented below.

- Describe the possible outcomes.
- Link each outcome to one or more random numbers.
- Choose a source of random numbers.
- Choose a random number.
- Based on the random number, note the "simulated" outcome.
- Repeat steps 4 and 5 multiple times; preferably, until the outcomes show a stable pattern.
- Analyze the simulated outcomes and report results.

**Note:** When it comes to choosing a source of random numbers
(Step 3 above), you have many options. Flipping a coin and
rolling dice are low-tech but effective. Tables of random numbers
(often found in the
appendices of statistics texts) are another option. And good
random number generators can be found on the internet.

## Random Number Generator

When you need random numbers, coin flipping, dice rolling, and random number tables can be cumbersome, particularly with large samples. As an alternative, use Stat Trek's Random Number Generator. With the Random Number Generator, you can select up to 1000 random numbers quickly and easily. The Random Number Generator is free. It can found in the Stat Trek main menu under the Stat Tools tab. Or you can tap the button below.

Random Number Generator## Simulation Example

In this section, we work through an example to show how to apply simulation methods to probability problems.

**Problem Description**

On average, suppose a baseball player hits a home run once in every 10 times at bat, and suppose he gets exactly two "at bats" in every game. Using simulation, estimate the likelihood that the player will hit 2 home runs in a single game.

**Solution**

Earlier we described seven steps required to produce a useful simulation. Let's apply those steps to this problem.

- Describe the possible outcomes. For this problem, there are two outcomes - the player hits a home run or he doesn't.
- Link each outcome to one or more random numbers. Since the player hits a home run in 10% of his at bats, 10% of the random numbers should represent a home run. For this problem, let's say that the digit "2" represents a home run and any other digit represents a different outcome.
- Choose a source of random numbers. For this problem, we used Stat Trek's Random Number Generator to produce a list of 500 two-digit numbers (see below).
- Choose a random number. The list below shows the random numbers that we generated.
- Based on the random number, note the "simulated" outcome. In this example, each 2-digit number represents two "at-bats" in a single game. Since the digit "2" represents a home run, the number "22" represents two home runs in a single game. Any other 2-digit number represents a failure to hit consecutive home runs in the game.
- Repeat steps 4 and 5 multiple times; preferably, until the outcomes show a stable pattern. In this example, the list of random numbers consists of 500 2-digit pairs; i.e., 500 repetitions of steps 4 and 5.
- Analyze the simulated outcomes and report results. In the list, we found 6 occurrences of "22", which are highlighted in red in the table. In this simulation, each occurrence of "22" represents a game in which the player hit consecutive home runs.

Random Numbers |

42 99 02 65 04 14 30 09 70 88 89 85 95 40 53 67 25 50 48 79 86 92 76 24 53 39 08 73 78 17 72 81 08 01 68 94 43 43 95 12 36 90 28 88 34 69 18 69 91 79 14 82 26 94 15 26 19 41 74 02 17 20 38 84 74 30 34 96 09 46 61 41 02 93 94 90 00 71 84 98 30 82 80 11 92 97 81 29 85 44 40 05 83 22 04 86 13 33 00 99 74 75 27 43 68 22 59 20 66 00 24 01 96 84 19 14 57 26 47 58 51 73 06 08 49 52 70 15 79 35 65 28 40 77 93 73 33 24 25 22 32 03 89 03 62 13 85 16 23 28 12 61 16 75 45 37 15 54 36 18 45 64 31 31 06 80 32 75 99 27 91 25 98 05 55 32 27 16 51 45 89 31 78 90 82 05 11 39 80 83 01 20 10 67 97 33 72 09 98 78 39 56 57 54 63 35 21 35 93 18 17 48 55 60 44 92 21 07 77 42 46 86 41 49 76 96 36 62 38 11 64 07 04 58 23 56 29 37 87 37 59 47 83 77 21 63 10 95 87 10 42 71 12 88 06 52 42 99 02 65 04 14 30 09 70 88 89 85 95 40 53 67 25 50 48 79 86 92 76 24 53 39 08 73 78 17 72 81 08 01 68 94 43 43 95 12 36 90 28 88 34 69 18 69 91 79 14 82 26 94 15 26 19 41 74 02 17 20 38 84 74 30 34 96 09 46 61 41 02 93 94 90 00 71 84 98 30 82 80 11 92 97 81 29 85 44 40 05 83 22 04 86 13 33 00 99 74 75 27 43 68 22 59 20 66 00 24 01 96 84 19 14 57 26 47 58 51 73 06 08 49 52 70 15 79 35 65 28 40 77 93 73 33 24 25 22 32 03 89 03 62 13 85 16 23 28 12 61 16 75 45 37 15 54 36 18 45 64 31 31 06 80 32 75 99 27 91 25 98 05 55 32 27 16 51 45 89 31 78 90 82 05 11 39 80 83 01 20 10 67 97 33 72 09 98 78 39 56 57 54 63 35 21 35 93 18 17 48 55 60 44 92 21 07 77 42 46 86 41 49 76 96 36 62 38 11 64 07 04 58 23 56 29 37 87 37 59 47 83 77 |

The simulation predicts that this particular player will hit consecutive home runs 6 times in 500 games. Thus, the simulation suggests that there is a 1.2% chance that he will hit two home runs in a single game. The actual probability, based on the multiplication rule, states that there is a 1.0% chance that this player will hit consecutive home runs in a game. While the simulation is not exact, it is very close. And, if we had generated a list with more random numbers, it likely would have been even closer.

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