Binomial distribution calculator
Type n and p and choose the probability you need. Fractions like 1/6 are fine. The answer is the exact sum of the binomial probabilities, never an approximation.
- Mean np
- 6.00
- Variance np(1 − p)
- 4.20
- Standard deviation
- 2.05
Show the working
X ~ B(20, 0.3): 20 independent trials, each a success with probability 0.3.
P(X = r) = 20Cr × 0.3r × 0.720 − r
P(X ≤ 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)
= 0.000798 + 0.00684 + 0.0278 + 0.0716 + 0.130 + 0.179 = 0.416
Mean E(X) = np = 6.00; variance Var(X) = np(1 − p) = 4.20.
Answers are rounded to 3 significant figures. A Level questions usually say how accurate to be: follow the question.
How to do it by hand
- Check the conditions: a fixed number of trials n, each a success or a failure, independent, with the same probability p each time.
- For one value, use P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ.
- For “at most”, add the probabilities from 0 up to r (your calculator's binomial CD does this). For “at least r”, use 1 − P(X ≤ r − 1).
Worked example
A basketball player scores 70% of her free throws, independently. She takes 12 free throws. Find the probability that she scores at least 10.
Solution
X ~ B(12, 0.7): 12 independent trials, each a success with probability 0.7.
P(X = r) = 12Cr × 0.7r × 0.312 − r
Add the probabilities from 10 up to 12, or use the complement: one minus the probabilities from 0 to 9.
P(X ≥ 10) = 1 − P(X ≤ 9) = 1 − 0.747 = 0.253
Mean E(X) = np = 8.40; variance Var(X) = np(1 − p) = 2.52.
Answer: P(X ≥ 10) = 0.253
Every number in this example is worked out by the same code as the tool above, and the code is tested against independent results from Python's SciPy library.
Where you need it
The binomial distribution and binomial hypothesis tests (critical regions) in A Level Statistics and IAL S1/S2; Poisson in Further Statistics and IAL S2.
Practise it: Statistical distributions · Hypothesis testing.
On your calculator
In the exam you use your own calculator, so practise it too. Our guides give the exact keystrokes for the Casio fx-CG50, TI-Nspire CX II, TI-84 Plus CE, Casio fx-991EX ClassWiz and Casio fx-991CW ClassWiz, with a worked example to check against:
Common mistakes
- Using P(X ≤ 10) for “fewer than 10”. “Fewer than 10” is P(X ≤ 9); with discrete distributions the end point matters.
- For “at least r”, working out 1 − P(X ≤ r) instead of 1 − P(X ≤ r − 1).
- Swapping n and p, or using the percentage (70) instead of the probability (0.7).
- Using the binomial when the trials are not independent, for example choosing without replacement from a small group.
Notes and practice
Questions students ask
When can I use the binomial distribution?
When there is a fixed number of independent trials, each with two outcomes (success or failure) and the same probability of success each time. Write X ~ B(n, p) and say what X counts.
What are the mean and variance of a binomial distribution?
The mean is np and the variance is np(1 − p). The tool shows both, with the standard deviation.
Why are P(X < 5) and P(X ≤ 5) different?
X can only be a whole number, so P(X < 5) stops at 4 while P(X ≤ 5) includes P(X = 5). Rewrite every inequality with ≤ before you use a cumulative function.