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Statistics · Statistical distributions

Chapter 1 · 4

The idea

Calculating binomial probabilities P(X = r)

The binomial probability formula P(X = r) = nCr · p^r · (1−p)^(n−r), why it has that shape (choose which r of the n trials succeed, then multiply the success and failure probabilities), how to evaluate it including the special cases r = 0 and r = n, and how it matches the binompdf function on a calculator.

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Statistics · Statistical distributions

Calculating binomial probabilities P(X = r)

The binomial probability formula P(X = r) = nCr · p^r · (1−p)^(n−r), why it has that shape (choose which r of the n trials succeed, then multiply the success and failure probabilities), how to evaluate it including the special cases r = 0 and r = n, and how it matches the binompdf function on a calculator.

Why it works

The formula

George throws at a target 1515 times, hitting with probability 0.480.48 each time. What is the chance of exactly 33 hits? The tempting answer, (0.48)3(0.48)^3, is nowhere close — it ignores that the other twelve throws must miss, and that the three hits could land anywhere among the fifteen. The binomial formula prices in all of it. For X∼B(n,p)X \sim B(n, p):

P(X=r)=(nr) pr (1−p)n−rP(X = r) = \binom{n}{r}\, p^{r}\, (1-p)^{n-r}

Why it has that shape

Picture one particular way to get rr successes — say the first rr trials succeed and the rest fail. That single sequence has probability

p×⋯×p⏟r successes×q×⋯×q⏟n−r failures=prq n−r,\underbrace{p \times \dots \times p}_{r \text{ successes}} \times \underbrace{q \times \dots \times q}_{n-r \text{ failures}} = p^{r} q^{\,n-r},

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