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Counting successes on a Galton board
Age 18 · Maths
Open to everyone until Wednesday, November 18
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The binomial distribution B(n, p) gives the number of successes X in n independent trials, where each trial succeeds with the same probability p. Its mean is np and its variance is np(1−p), so outcomes cluster around np and grow rarer the further they lie from it.
In the picture
- The funnel: each ball released runs one full set of n trials
- A peg: one Bernoulli trial, bouncing right (success, probability p) or left (failure, probability 1−p)
- The middle bin: the tallest heap sits at the mean np
- An outer bin: its shallow heap shows how far results spread, measured by the variance np(1−p)
Examples
- Flipping a fair coin 10 times and counting heads: X ~ B(10, 0.5), with mean 5 and variance 2.5.
- A basketball player who makes 70% of free throws takes 20 shots: on average they make 14, with variance 4.2.
Key takeaways
- For X ~ B(n, p): mean np, variance np(1−p), and P(X = k) = C(n, k) p^k (1−p)^(n−k).
- The model only works if the trials are independent and every trial has the same probability p.
Quiz: Counting successes on a Galton board (Age 18)
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A variable X follows B(20, 0.3). What are its mean and variance?
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In which of these situations does the binomial model NOT apply?
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For X ~ B(n, p), around which value do the outcomes cluster?
The answers, each with its explanation, are in the app. This quiz is free there while the lesson is in the free week.
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