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Probability
Simulations

Learn to design and run simulations that estimate probability, then interpret results to make predictions about real-world events.

Pre-Algebra

What Is Probability?

Probability measures how likely an event is to happen. It is always a number between \(0\) and \(1\), where \(0\) means impossible and \(1\) means certain.

For example, the probability of flipping heads on a fair coin is \(\displaystyle P(\text{heads}) = \frac{1}{2} = 0.5\).

Theoretical vs. Experimental Probability

Theoretical
What should happen based on math. Uses the formula \(\frac{\text{favorable}}{\text{total}}\). No experiment needed.
Experimental
What actually happens when you run trials. Uses the formula \(\frac{\text{times event occurred}}{\text{total trials}}\).

What Is a Simulation?

A simulation is an experiment that models a real-world situation using random devices: coins, dice, spinners, random number generators, or cards.

Why Simulate?
When a real experiment is too expensive, time-consuming, or dangerous to repeat many times.
Key Idea
The simulation device must have the same probability as the real event it represents.

Matching Devices to Probabilities

Choose a random device whose equally likely outcomes match the probability of the event you are modeling.

Real-World Event Probability Simulation Device
Free throw made \(\frac{1}{2}\) Coin flip (H = made)
Rain tomorrow \(\frac{1}{3}\) Die: \(1\)-\(2\) = rain, \(3\)-\(6\) = no rain
Quiz answer correct \(\frac{1}{4}\) 4 cards: draw the ace
Winning lottery \(\frac{1}{100}\) Random number \(1\)-\(100\)

The Law of Large Numbers

As you increase the number of trials in a simulation, the experimental probability approaches the theoretical probability.

\(P=0.5\) 0 0.5 1.0 0 50 200 500 1000 Number of Trials Experimental probability

Simulation vs. Prediction

Simulation
Uses a random device to model an event many times. Gives an estimate of probability based on data you collect.
Prediction
Uses a known probability to forecast future outcomes. Formula: \(\text{Expected} = P \times n\) where \(n\) is the number of trials.

How to Design a Simulation

1
Identify the event and its theoretical probability. What real situation are you modeling?
2
Choose a device whose outcomes match the probability. Make sure each outcome is equally likely.
3
Define a trial. Decide what one complete "run" of the experiment looks like and what counts as a success.
4
Run many trials (at least 20-50) and record your results in a table or tally chart.
5
Calculate experimental probability: \(\displaystyle P_{\text{exp}} = \frac{\text{successes}}{\text{total trials}}\)

Example 1: Coin Flip Simulation

Problem: A basketball player makes \(50\%\) of her free throws. Simulate \(10\) free throws using a coin. Heads = made, tails = missed.

H T H H T T H H T H Trial 1 through Trial 10

Example 2: Die Roll Simulation

Problem: There is a \(\frac{1}{3}\) chance of rain each day this week. Use a die to simulate \(7\) days. Roll \(1\)-\(2\) = rain, roll \(3\)-\(6\) = no rain.

2 Rain 5 Dry 1 Rain 4 Dry 6 Dry 2 Rain 3 Dry Day 1 through Day 7

Example 3: Random Numbers

Problem: A cereal box has a \(\frac{1}{5}\) chance of containing a prize. Use random numbers \(1\) through \(10\) to simulate buying \(20\) boxes. Numbers \(1\)-\(2\) = prize.

7 1 9 4 6 2 8 5 3 10 5 3 2 7 1 8 6 9 2 4 = Prize (\(1\) or \(2\)) = No Prize (\(3\)-\(10\)) 20 simulated box purchases

Interpreting Results

After running a simulation, you need to interpret what your data tells you. Ask these key questions:

1
What is the experimental probability? Divide successes by total trials.
2
How does it compare to theoretical? Small differences are normal. Large differences suggest too few trials.
3
Can you make a prediction? Multiply the experimental probability by a future number of trials: \(\text{Expected} = P_{\text{exp}} \times n\).

Making Predictions from Data

Scenario: In a simulation of \(50\) trials, a spinner landed on blue \(15\) times. Predict how many times it will land on blue in \(200\) spins.

1
Find experimental probability: \(\displaystyle P_{\text{exp}} = \frac{15}{50} = 0.3\)
2
Multiply by the new number of trials: \(0.3 \times 200 = 60\)
3
Interpret: You can predict blue will land about 60 times in 200 spins.

Common Mistakes to Avoid

✕ Wrong Device
Using a coin (probability \(\frac{1}{2}\)) to model an event with probability \(\frac{1}{3}\). The device must match the probability.
✕ Too Few Trials
Running only \(5\) trials and treating the result as definitive. More trials give more reliable estimates.
✕ Expecting Exact Match
Thinking experimental probability must equal theoretical. Some variation is completely normal.

Summary of Key Concepts

Concept Definition Formula
Theoretical Probability What should happen mathematically \(\frac{\text{favorable}}{\text{total}}\)
Experimental Probability What actually happened in trials \(\frac{\text{successes}}{\text{trials}}\)
Simulation Model using random devices Run many trials, record data
Prediction Forecast using known probability \(P \times n\)
Law of Large Numbers More trials = closer to theoretical \(P_{\text{exp}} \to P_{\text{theo}}\)

Question 1 of 5

A spinner has a \(\frac{1}{4}\) chance of landing on red. Which device best simulates this event?

Question 2 of 5

In a simulation of \(40\) coin flips, heads came up \(18\) times. What is the experimental probability of heads?

Question 3 of 5

A simulation found that an event occurred \(12\) times in \(60\) trials. If you run \(300\) more trials, about how many times would you expect the event to occur?

Question 4 of 5

Why does increasing the number of trials in a simulation generally give a better estimate of the true probability?

Question 5 of 5

A student wants to simulate an event with probability \(\frac{1}{3}\). They flip a coin and say "heads = event happens." What is the problem with this simulation?