Probability lab
Set success probability and trial count, then compare theory with simulation. Ten lessons take you from simple events to expectation and independence.
Independent trials simulator
Each click starts a fresh series. A pseudorandom simulation illustrates the model; it does not predict lottery results.
How do I use this page?
- Read a lesson and its worked example.
- Enter an answer, check it and compare the solution steps.
- Generate another exercise or take the 10-question test.
What this page does not do: predict future lottery outcomes.
Course: 10 lessons
- Events and probability
- The complement
- Independent trials
- At least one success
- Unordered combinations
- Without replacement
- Expected successes
- Net expected value
- Variance of successes
- Streaks and the gambler’s fallacy
For non-terminating decimals, round to three decimal places or enter a fraction such as 1/3.
Optional local storage only. No account or server submission. Turn off to remove saved progress.
1. Events and probability
For equally likely outcomes, P(A) is the number of favourable outcomes divided by the total. It lies between 0 and 1.
Worked example
3 red balls out of 10: P = 3/10 = 0.3 = 30%.
Watch out
Do not divide by the number of balls of the other colour. Check that every ball is equally likely.
2. The complement
An event and its complement cover all possibilities and cannot occur together. P(not A) = 1 − P(A).
Worked example
A rain probability of 0.2 means a no-rain probability of 0.8 in the same model.
Watch out
The complement is everything outside A, not an arbitrary alternative event.
3. Independent trials
Multiply probabilities for independent events. A fair coin has P(heads) = 1/2 on each independent toss.
Worked example
Three heads: 1/2 × 1/2 × 1/2 = 1/8.
Watch out
Do not assume independence. Drawing without replacement changes the pool.
4. At least one success
Calculate no successes and subtract from 1. For n independent trials with fixed p: 1 − (1 − p)^n.
Worked example
Two fair tosses: 1 − (1/2)² = 3/4.
Watch out
At least one is not exactly one. Two successes also qualify.
5. Unordered combinations
The pair A,B is the same selection as B,A. There are n(n−1)/2 pairs from n distinct items.
Worked example
From 5 items you can select 5 × 4 / 2 = 10 pairs.
Watch out
Do not divide by 2 if order matters, such as selecting a chair and deputy.
6. Without replacement
After drawing a red ball, both the red count and total count decrease by one. Multiply the first probability by the conditional second probability.
Worked example
4 red out of 10: two reds have probability 4/10 × 3/9 = 2/15.
Watch out
Do not reuse 4/10 for the second draw. That would be replacement.
7. Expected successes
In n trials with success probability p, the expected count is np. It need not be an integer.
Worked example
5 fair tosses average 2.5 heads across repeated series.
Watch out
Expectation is not a promise for one series.
8. Net expected value
Multiply each payout by its probability, add the results and subtract the cost. Use one currency throughout.
Worked example
Payout 4 with probability 1/4, cost 2: 4 × 1/4 − 2 = −1.
Watch out
Do not confuse payout with profit. Refunding the ticket price gives zero profit.
9. Variance of successes
For independent Bernoulli trials, the variance of the success count is np(1−p). Standard deviation is its square root.
Worked example
8 fair tosses: variance 8 × 1/2 × 1/2 = 2.
Watch out
Variance describes spread, not a fault in the random device.
10. Streaks and the gambler’s fallacy
Past independent outcomes do not change the next probability. A long streak can look unusual but creates no debt to repay.
Worked example
After five heads, the next independent fair toss still has P(heads) = 1/2.
Watch out
Do not treat history as a guarantee. Simulation illustrates variation, not prediction.
Final test: 10 questions
One new question from each lesson. No time limit. Answers and explanations appear after the test. You may stop and restart.
Further reading
Original exercises by Pewniaki. The mental-maths identities are explained algebraically; no claim of historical origin or guaranteed speed advantage is made.