Binomial distribution
Practice Flashcards
When is X binomial, X ~ B(n, p)?
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All Flashcards in Topic 4.8
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4.8.19 cards
When is X binomial, X ~ B(n, p)?
Fixed number n of independent trials, two outcomes each, with constant success probability p.
What does binompdf(n, p, k) give?
P(X = k) — the probability of exactly k successes.
What does binomcdf(n, p, k) give?
P(X ≤ k) — the probability of at most k successes.
State the binomial probability formula.
P(X = k) = ⁿCₖ pᵏ (1−p)ⁿ⁻ᵏ.
How do you find P(X ≥ k)?
1 − P(X ≤ k − 1) = 1 − binomcdf(n, p, k − 1).
How do you find P(a ≤ X ≤ b)?
binomcdf(n, p, b) − binomcdf(n, p, a − 1).
How do you find P(at least one)?
1 − P(X = 0).
Why might a 'without replacement' situation not be binomial?
The probability of success changes between trials, so p is not constant.
Finding n for 'at least one' — round up or down?
Round up, since you need to reach the target probability with a whole number of trials.
4.8.29 cards
What is the mean of X ~ B(n, p)?
E(X) = np.
What is the variance of X ~ B(n, p)?
Var(X) = np(1 − p).
What is the standard deviation of a binomial?
√(np(1 − p)).
What does the binomial mean represent?
The expected number of successes in n trials.
Common slip in the variance?
Using np or np² instead of np(1 − p) — you must multiply by both p and (1 − p).
How do you find p from the mean and variance?
variance ÷ mean = 1 − p, so p = 1 − variance/mean.
How do you then find n?
n = mean ÷ p.
Are np and np(1 − p) in the formula booklet?
Yes — both the binomial mean and variance are given.
X ~ B(50, 0.2): mean and variance?
Mean 10, variance 8.
Topic 4.8 study notes
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