Expected number of successes: For X ~ B(n, p) the mean (expected number of successes) is E(X) = np — just trials × success probability.
IB-style question — the mean
X ~ B(50, 0.2). Find the expected number of successes.
Step by step
- Mean = np.
Final answer
E(X) = 10.
This is the 'expected number': A 'how many do you expect to…' binomial question is just np.
Spread of the binomial: The variance is Var(X) = np(1 − p), and the standard deviation is its square root, √(np(1−p)).
IB-style question — variance & sd
X ~ B(50, 0.2). Find the variance and standard deviation.
Step by step
- Variance = np(1 − p).
- Standard deviation = √variance.
Final answer
Variance = 8; standard deviation ≈ 2.83.
Use (1 − p), not p, twice: Variance is np(1 − p) — multiply by both p and (1 − p); a common slip is np² or n p.
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Use mean and variance together to find n and p: Given the mean np and variance np(1−p), divide them: variance ÷ mean = (1 − p), which gives p; then n = mean ÷ p.
IB-style question — find n and p
A binomial variable has mean 24 and variance 14.4. Find n and p.
Step by step
- variance ÷ mean = (1 − p).
- n = mean ÷ p.
Final answer
p = 0.4 and n = 60.
Divide to remove n: Dividing variance by mean cancels n, leaving (1 − p) — a clean way in.