Free maths tool · Level: Ages 17 to 18
Binomial distribution
Set the number of attempts, the chance of each one and the number of successes. The lit bar is the probability you are after, and the formula below shows how it is built.
The binomial distribution gives the probability of exactly k successes in n independent attempts that each have the same probability p. The formula is P(X = k) = C(n,k) · pᵏ · (1-p)ⁿ⁻ᵏ. C(n,k) counts the orders those k successes can fall in, and the rest is the probability of any one such order.
Where C(n,k) comes from
Draw the same experiment as a probability tree and every path with k successes has exactly the same probability: k lots of p and the rest lots of 1-p. They differ only in order.
So you work that probability out once and then count the paths. That count is C(n,k), and it is nothing more than a count. Three attempts with exactly one success give three paths, and C(3,1) is indeed 3.
At most and at least
A question like "at most 3 times" is not about one bar but about every bar up to and including 3. Switch the cumulative layer on and they are coloured together, with their total beside them.
For "at least 4 times" you reuse that answer: anything that is not at most 3 is at least 4, so 1 minus the cumulative probability. That saves a lot of adding once n gets large.
When the normal distribution can step in
Turn n up and the bars form a bell shape. That is no accident, and it is why a binomial probability can be approximated by a normal distribution with mean n·p and standard deviation the square root of n·p·(1-p).
The rule of thumb is that n·p and n·(1-p) are both at least 5. Set p to 0.05 and keep n small to see why: the distribution bunches against the left edge while the bell curve stays symmetric, so it misses. The tool draws the curve either way and says whether the rule holds.
Are probability distributions where it stalls?
A tutor works through your exercises step by step, from the formula to the calculator.
Frequently asked questions
- When do you use the binomial distribution?
- When the same experiment is repeated a fixed number of times, each attempt has only two outcomes, the probability stays the same throughout, and the attempts do not affect one another. Drawing without replacement is out, because p changes along the way.
- What is the binomial distribution formula?
- P(X = k) = C(n,k) · pᵏ · (1-p)ⁿ⁻ᵏ. C(n,k) counts the orders, pᵏ covers the k successes and (1-p)ⁿ⁻ᵏ covers the failures that are left.
- How do you calculate P(X ≤ k)?
- Add the probabilities from k = 0 up to and including your k. A graphing calculator does it in one step with binomcdf; by hand you add the bars. For P(X ≥ k), take 1 minus P(X ≤ k-1).
- What are the expected value and standard deviation?
- The expected value is n·p and the standard deviation is the square root of n·p·(1-p). Over 20 tosses of a fair coin you expect 10 heads, with a spread of about 2.24.
- When can you approximate the binomial with the normal distribution?
- When n·p and n·(1-p) are both at least 5. The distribution is then symmetric enough for the bell curve to sit close. If p is very small and n is modest, the approximation fails and you should stay binomial.