7.5 Binomial Probabilities

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Notes

The probability of exactly $k$ successes in $n$ trials is $P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}$, where $\binom{n}{k}$ counts the ways to arrange the successes.

Useful values: $\binom{3}{1} = 3$, $\binom{3}{2} = 3$, $\binom{4}{2} = 6$.
For "at least one," use the complement: $P(X \ge 1) = 1 - P(X = 0)$.
Example: $n = 3$, $p = 0.5$: $P(X = 2) = 3(0.5)^2(0.5) = 0.375$.

Practice

A fair coin is flipped 4 times. Given $\binom{4}{2} = 6$, the probability of exactly 2 heads is:

A binomial variable has $n = 3$ and $p = 0.2$. What is $P(X = 0)$?

Each of 3 independent attempts succeeds with probability $0.1$. What is the probability of at least one success?

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