Probability Simulations — I can design and run a simulation to estimate probability and interpret results. - Pre Algebra

SECTION 1

Where Did Probability Simulations Come From?

Have you ever wondered how weather forecasters predict a 70 % chance of rain? They can't flip a coin to figure that out. Instead, they run simulations — experiments that copy a real-world situation — over and over again to see what happens. This idea of using repeated experiments to estimate chance has a fascinating history.

1654 • The Gambling Letters

Two French mathematicians, Blaise Pascal and Pierre de Fermat, wrote letters to each other about dice games. Their work created the first rules of probability.

1777 • Buffon's Needle

The Comte de Buffon dropped a needle on lined paper thousands of times. By counting how often it crossed a line, he estimated the value of π. This was one of the earliest probability simulations.

1946 • The Monte Carlo Method

Scientists Stanislaw Ulam and John von Neumann used random number simulations to solve nuclear physics problems. They named the method after the famous Monte Carlo casino.

Today • Simulations Everywhere

Computers now run millions of simulations in seconds. Video game designers, doctors, and sports analysts all use simulations to predict outcomes.

Here is the big question these thinkers tried to answer: When the math is too hard or the real situation is too complicated, can we just try it many times and see what happens? The answer is yes — and that is exactly what a probability simulation does.

SECTION 2

Core Principles of Probability Simulations

Before you build a simulation, you need to understand a few key ideas. These ideas form the foundation for every simulation you will ever run.

Trial

A trial is one run of your experiment. If you flip a coin once, that is one trial. If you roll a die once, that is one trial.

Outcome

An outcome is the result of a single trial. When you flip a coin, the outcome is heads or tails.

Experimental Probability

Experimental probability is the probability you calculate from your simulation data. It equals the number of times an event happened divided by the total number of trials.

Theoretical Probability

Theoretical probability is the "perfect" probability you calculate using math, without doing any experiments. For example, the theoretical probability of heads on a fair coin is 1/2.

Law of Large Numbers

The Law of Large Numbers says that the more trials you run, the closer your experimental probability will get to the theoretical probability.

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KEY TAKEAWAY

Think of a simulation like taste-testing soup while cooking. One tiny spoonful might taste too salty or not salty enough. But if you stir the pot and try many spoonfuls, you get a much better idea of the true flavor. In a simulation, each trial is a "spoonful," and the more you try, the better your estimate becomes.

SECTION 3

Seeing a Simulation in Action

Let's look at a picture of what happens when you simulate flipping a fair coin. The diagram below shows the experimental probability of heads after different numbers of flips. Notice how the line is jumpy at first but gets closer and closer to 0.50 (the theoretical probability) as more flips happen.

With only 5 flips, the experimental probability (0.80) is far from the true probability (0.50). By 200 flips, the value settles very close to 0.50. This is the Law of Large Numbers in action.

Look at the first dot on the left. After only 5 flips, 4 of them were heads, so the experimental probability was 0.80. That feels way off, right? But that is totally normal for a small number of trials. As you move to the right and add more flips, the line hugs the dashed yellow line at 0.50 more and more tightly. This is why running many trials matters.

SECTION 4

The Math Behind Simulations

You only need one formula to turn your simulation data into a probability estimate. Let's break it down.

EXPERIMENTAL PROBABILITY

P(event) = Number of times the event happens ÷ Total number of trials

P(event) means "the probability of the event." For example, if you roll a die 50 times and get a 6 exactly 9 times, then P(rolling a 6) = 9 ÷ 50 = 0.18.

THEORETICAL PROBABILITY (for comparison)

P(event) = Number of favorable outcomes ÷ Total number of possible outcomes

A standard die has 6 faces, so the theoretical probability of rolling a 6 is 1 ÷ 6 ≈ 0.167. Your experimental result (0.18) is close — but not exact. That is perfectly okay!

When you design a simulation, you pick a model (a simple device that copies the real situation). Here are some common models you can use.

Real-World Situation	Good Model	Why It Works
50/50 chance (yes or no)	Flip a coin	Two equally likely outcomes
1 out of 6 chance	Roll a die	Six equally likely outcomes
1 out of 10 chance	Draw a digit 0–9 from a bag	Ten equally likely outcomes
3 out of 4 chance (75 %)	Spin a spinner with 3 colored and 1 white section	Three favorable out of four equal sections

Matching real situations to simulation models

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Quick Tip

A random number generator (on a calculator or computer) can replace any of these models. For a 50/50 event, let odd numbers = yes and even numbers = no.

SECTION 5

How to Design a Simulation Step by Step

Every simulation follows the same five-step process. The flowchart below shows the steps, and then we'll dig into each one.

Follow these five steps every time you set up a simulation. The notes on the sides show quick examples for a coin-flip situation.

Step 1 – State the question. Write down exactly what probability you want to estimate. Example: "What is the probability of getting at least 2 heads in 3 coin flips?"
Step 2 – Choose a model. Pick a random device that matches the situation. A fair coin works for anything with two equally likely outcomes.
Step 3 – Define one trial and "success." One trial = flip the coin 3 times. Success = 2 or more heads.
Step 4 – Run many trials. Repeat the trial at least 30 times (more is better) and record each outcome.
Step 5 – Calculate and interpret. Divide the number of successes by the total trials. Compare to the theoretical probability if you know it.

SECTION 6

Worked Example: Free-Throw Simulation

A basketball player makes about 60 % of her free throws. Her coach wants to estimate the probability that she makes at least 4 out of 5 free throws in a row. Let's design and run a simulation.

Free-Throw Simulation

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Step 1 — State the Question

What is the probability that the player makes at least 4 out of 5 free throws?
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Step 2 — Choose a Model

The player makes 60 % of her shots. We can use random digits 0–9. Let digits 0, 1, 2, 3, 4, 5 (six digits = 60 %) represent a made shot. Let digits 6, 7, 8, 9 (four digits = 40 %) represent a missed shot.
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Step 3 — Define One Trial

One trial = pick 5 random digits. A "success" means 4 or 5 of those digits are in the range 0–5.
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Step 4 — Run 20 Trials

Here are 20 trials using a random number table. Each group of 5 digits is one trial. We underline digits 0–5 (made shots). Trial 1: 3 8 1 0 4 → 4 made ✓ | Trial 2: 7 2 9 6 1 → 2 made ✗ | Trial 3: 5 5 0 3 8 → 4 made ✓ | Trial 4: 6 4 1 7 9 → 2 made ✗ | Trial 5: 2 0 3 5 4 → 5 made ✓ | Trial 6: 8 7 6 9 1 → 1 made ✗ | Trial 7: 4 3 2 0 5 → 5 made ✓ | Trial 8: 9 8 7 6 3 → 1 made ✗ | Trial 9: 1 5 4 2 7 → 4 made ✓ | Trial 10: 0 3 8 5 1 → 4 made ✓ | Trial 11: 6 7 2 9 8 → 1 made ✗ | Trial 12: 3 4 0 1 2 → 5 made ✓ | Trial 13: 5 9 3 7 0 → 3 made ✗ | Trial 14: 1 4 5 2 6 → 4 made ✓ | Trial 15: 8 0 9 3 7 → 2 made ✗ | Trial 16: 2 5 4 1 0 → 5 made ✓ | Trial 17: 7 6 8 9 4 → 1 made ✗ | Trial 18: 3 0 5 2 1 → 5 made ✓ | Trial 19: 9 4 7 8 6 → 1 made ✗ | Trial 20: 0 1 5 3 8 → 4 made ✓
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Step 5 — Calculate and Interpret

Count the successes (trials with 4 or 5 made): Trials 1, 3, 5, 7, 9, 10, 12, 14, 16, 18, 20 = 11 successes out of 20 trials.

Experimental probability = 11 ÷ 20 = 0.55 or 55 %. Based on this simulation, there is about a 55 % chance the player makes at least 4 of her 5 free throws. With more trials, this estimate would become even more accurate.

SECTION 7

Strengths and Limitations of Simulations

Simulations are powerful, but they aren't perfect. Let's compare their strengths and weaknesses so you know when to use them — and when to be careful.

Strengths ✅	Limitations ⚠️
Can estimate probabilities that are hard to calculate with formulas.	Results are estimates, not exact answers.
Easy to set up with coins, dice, cards, or a calculator.	A small number of trials can give misleading results.
You can model complicated real-life situations.	The model must truly match the real situation, or the results will be off.
More trials always improve accuracy.	Running many trials by hand takes time.

Strengths vs. Limitations of Probability Simulations

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KEY TAKEAWAY

A simulation is like practicing a play in sports before the big game. It won't tell you the exact final score, but it gives you a really good feel for what might happen. The more you practice, the better your prediction.

SECTION 8

Connecting to Advanced Probability

What you're learning now — running simulations by hand — is the same idea used in high-powered computing. The table below shows how your middle-school simulation connects to what professionals use.

What You Do Now	What Professionals Do
Flip coins or roll dice for 20–50 trials.	Run computer programs for millions of trials.
Use digits 0–9 from a random number table.	Use algorithms that generate truly random numbers.
Estimate simple probabilities (coin flips, die rolls).	Estimate hurricane paths, stock prices, and disease spread.
Compare to theoretical probability to check your answer.	Use statistics to measure how confident they are in the result.

Your simulation skills today vs. professional Monte Carlo methods

In high school and college, you'll learn about the Monte Carlo method and use coding languages like Python to run simulations with millions of trials. The core idea, though, is exactly what you are doing right now: repeat, count, and divide.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

Why does running more trials in a simulation give you a better estimate of the true probability?

Show answer

The Law of Large Numbers tells us that as we increase the number of trials, the experimental probability gets closer and closer to the theoretical probability. With only a few trials, random chance can make the results look very different from the true probability. But as we run more and more trials, the lucky and unlucky results start to balance each other out, giving us a more reliable estimate.

Question 2: BASIC CALCULATION

A student rolls a die 40 times to simulate a game. She is looking for the probability of rolling a number less than 3 (that means rolling a 1 or 2). She gets a 1 or 2 exactly 15 times. What is the experimental probability? What is the theoretical probability?

Show answer

Experimental probability = 15 ÷ 40 = 0.375 or 37.5 %. The theoretical probability of rolling less than 3 on a die is 2 favorable outcomes ÷ 6 total outcomes = 2/6 ≈ 0.333 or about 33.3 %. The experimental probability (37.5 %) is a bit higher than the theoretical probability (33.3 %). This small difference is normal for just 40 trials.

Question 3: INTERMEDIATE

You want to estimate the probability of getting exactly 2 tails when you flip a coin 3 times. Describe how you would set up a simulation. What would be one trial? What counts as a success? How many trials would you recommend?

Show answer

Model: use a fair coin (or use random digits where odd = heads and even = tails). One trial means flipping the coin 3 times and writing down how many tails you get. A "success" is getting exactly 2 tails out of the 3 flips. You should run at least 30 to 50 trials for a decent estimate, but 100 or more would be even better. After recording all trials, divide the number of successes by the total trials to find the experimental probability.

Question 4: APPLIED

A cereal company puts one of 4 different toy figures inside each box. The figures are equally likely. You want to collect all 4 figures. Design a simulation to estimate the probability that you will collect all 4 figures within 8 boxes of cereal.

Show answer

Model: use a die (ignore rolls of 5 and 6 and re-roll, so you only get 1–4) or use random digits 1–4. One trial = pick 8 random numbers from 1 to 4 (each number represents the figure in one cereal box). A success happens when all four numbers (1, 2, 3, and 4) appear at least once in your 8 picks. Run at least 30 trials. Count how many of those trials are successes, then divide by the total number of trials. This gives your estimated probability of collecting all 4 figures in 8 boxes.

Question 5: CRITICAL THINKING

Two students both simulated the probability of rain on a given day (the true probability is 30 %). Student A ran 10 trials and got 5 rainy days (50 %). Student B ran 200 trials and got 64 rainy days (32 %). Which student's estimate is more trustworthy and why? Could Student A's result happen even though the coin was fair? Explain.

Show answer

Student B's estimate (32 %) is much more trustworthy because it is based on 200 trials, while Student A used only 10 trials. By the Law of Large Numbers, more trials bring the experimental probability closer to the true value. Student A's result of 50 % could absolutely happen with just 10 trials even though the true probability is 30 %. With so few trials, random variation is large. It does not mean anything went wrong — it simply means the sample was too small to be reliable. This is exactly why we recommend running as many trials as possible.

SUMMARY

Pulling It All Together

Summary

A probability simulation is an experiment you design to estimate how likely an event is. You start by picking a model (like a coin, die, or random number generator) that matches the real-world situation. Then you define one trial and decide what counts as a success. After running many trials, you calculate the experimental probability by dividing the number of successes by the total number of trials.

The Law of Large Numbers guarantees that more trials give a better estimate. Your experimental probability will get closer to the theoretical probability as the number of trials grows. Simulations are especially useful when the math for the theoretical answer is too complex. This same idea — repeat, count, divide — powers the Monte Carlo methods used by scientists and engineers around the world.