Equations · Probability and statistics
Binomial distribution
The probability of exactly k successes in n independent tries, each with success probability p.
P(X = k) = C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ
Aethel says
the probability that X equals k, equals n choose k, times p to the power of k, times the quantity 1 minus p, to the power of n minus k
What it says
Any particular sequence with k successes and n − k failures has probability p to the k times 1 − p to the n − k. The binomial coefficient counts how many such sequences there are.
Ten fair coin tosses give exactly five heads with probability 252/1024, about 25%.
Symbol by symbol
| Symbol | Say it | Means |
|---|---|---|
| n | n | the number of independent tries |
| k | k | the number of successes |
| p | p | the probability of success on each try |
| C(n, k) | n choose k | the number of ways to place k successes among n tries |
More on!Factorial
Where you meet it
Probability and statistics courses, quality control and clinical trials.
Questions
How do you say Binomial distribution out loud?
P(X = k) = C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ is read: “the probability that X equals k, equals n choose k, times p to the power of k, times the quantity 1 minus p, to the power of n minus k.” That is exactly what Aethel says when it meets the formula in a book.
What does n mean in Binomial distribution?
n (said “n”) is the number of independent tries.
What does k mean in Binomial distribution?
k (said “k”) is the number of successes.
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