Coding Grandmaster · project 23 of 46 · premium, free for now

The binomial distribution

Write binomial(n, k, p) and use it to print the chance of exactly 3 heads in 5 fair flips, the chance of at least 4 heads, and the total of all the chances (rounded to 6 places).

Learn

Flip a coin 5 times: what is the chance of exactly 3 heads? There are comb(5, 3) = 10 ways to choose which flips are heads, and each such sequence has probability p³ × (1 − p)². So P(k) = comb(n, k) × p^k × (1 − p)^(n − k). Adding P(k) for every k from 0 to n gives 1, because something must happen. The long-run average is n × p.

Your task

Write binomial(n, k, p) and use it to print the chance of exactly 3 heads in 5 fair flips, the chance of at least 4 heads, and the total of all the chances (rounded to 6 places).

Stretch challenge

Find the expected number of heads in 100 flips and the chance of exactly 50.

← Bayes and the medical test How computers fake randomness →

Python

Press Run to see what your program prints.

Keep going in Coding Grandmaster

Counting outcomes

Write ways(target), the number of dice pairs that add up to target, and as_fraction(top, bottom) that returns text like 1/6 in lowest terms. Print the chance of a total of 7.

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The OR rule

Build the 52-card deck as (rank, suit) pairs, then count the cards that are the chosen suit, the chosen rank, both, and either. Print the four counts and the chance of either as a fraction in lowest terms.

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Given that…

For the chosen club print how many students are in it, how many passed, and P(passed | club) as a percentage with one decimal place. Then print P(passed) for all students.

Start →22

Bayes and the medical test

Write posterior(prior, sensitivity, specificity) returning P(sick | positive) rounded to 4 decimal places. Print it for prior 0.01, sensitivity 0.9, specificity 0.9.

Start →24

How computers fake randomness

Write next_value(seed) with that formula, and roll(seed) = 1 + seed % 6. Starting from the given seed, roll 600 times (feeding each result back in) and print how many times each face 1 to 6 came up.

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Practise the idea

🎯 Play Plinko Probability Lab

Drop balls through a peg board and see how gravity and chance shape the results.

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FAQ

The binomial distribution: common questions

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