Probability Simulator

Connect a formula to observable experiments

A placement aptitude test contains ten independent true/false questions. If a student guesses randomly, what is the probability of exactly seven correct answers? We can calculate the answer using the binomial distribution and also estimate it by simulating many ten-question attempts.

The two answers serve different purposes. Theory describes the model exactly. Simulation demonstrates long-run frequency, helps test implementation and supports cases where an exact formula may later be unavailable.

Four conditions must hold

1. A fixed number n of trials occurs in every experiment.
2. Each trial has two modeled outcomes: success or failure.
3. The success probability p remains constant.
4. Trials are independent under the model.

Let X be the number of successes. Then X can take integer values 0 through n. Its expected value is E[X]=np and variance is np(1−p).

Choose positions, then assign outcomes

P(X=k) = C(n,k) × p^k × (1−p)^(n−k)

C(n,k) counts which k of the n positions are successes. Every particular arrangement has probability p^k(1−p)^(n−k). Multiplication combines independent outcomes; multiplication by the combination count covers all arrangements with exactly k successes.

At least k successes

P(X≥k) = Σ from j=k to n of P(X=j)

“Exactly seven” and “at least seven” are not interchangeable. At least seven includes 7, 8, 9 and 10. For thresholds near zero, a complementary calculation 1−P(X<k) may require fewer terms.

Turn probability into repeated Boolean trials

1. Generate a uniform pseudo-random number u in [0,1).
2. Count success when u<p.
3. Repeat n times to obtain one value of X.
4. Record whether X=k and whether X≥k.
5. Repeat the complete experiment M times.
6. Estimate probability as matching count divided by M.

estimated P(X=k) = number of experiments with k successes / M

A pseudo-random generator is deterministic. A seed selects its starting state. Using the same program, inputs and seed gives the same simulation, which is essential for debugging and teaching. Changing the seed provides another valid sample.

Exactly seven correct guesses out of ten

For random guessing on independent true/false questions, n=10, p=0.5 and k=7.

1. C(10,7)=120.
2. 0.5^7 × 0.5^3 = 0.5^10 = 1/1024.
3. P(X=7)=120/1024=0.1171875.
4. P(X≥7)=[C(10,7)+C(10,8)+C(10,9)+C(10,10)]/1024.
5. The numerator is 120+45+10+1=176, so the result is 0.171875.

A run of 10,000 experiments might produce 0.119 or 0.115 for exactly seven. That difference is expected sampling variation, not automatic evidence that the formula or code is wrong.

Complete Python implementation

The program accepts a general binomial experiment, calculates exact probabilities, runs a reproducible simulation and compares both exact-k and at-least-k events. It also checks the simulated average successes against np.

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Trace one experiment with n=4 and p=0.5

Start with probabilities that must be exact

Convergence is a tendency, not equality

By the law of large numbers, relative frequency tends toward the modeled probability as M grows. Typical Monte Carlo error decreases proportionally to 1/√M. To reduce typical error by a factor of ten, we therefore need roughly one hundred times as many experiments.

Simulation output is evidence about the code and the assumed process, not proof that real-world trials satisfy independence or constant p. Model validation must come before numerical confidence.

Check theoretical and experimental reasoning

Extensions

1. Display the complete theoretical and simulated distribution for k=0…n.
2. Repeat many seeds and summarize the distribution of estimation error.
3. Add a confidence interval for the estimated event probability.
4. Simulate unequal trial probabilities and explain why the binomial formula no longer applies.
5. Build a without-replacement simulation and compare it with a hypergeometric model.

Probability is a model; simulation is an experiment on that model

The exact binomial result and Monte Carlo estimate should be connected but not confused. Careful work states assumptions, derives the event probability, records the seed and sample size, measures error and resists interpreting random variation as a contradiction.