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Binomial probability

A binomial situation has a fixed number of independent trials, and each trial has two outcomes. The probability of exactly $k$ successes is the number of orders times the probability of one such order. Two dice throws are independent, so the probabilities are multiplied. The same model is used for a coin tossed several times, when the probability is the same on every toss.

Try it yourself

The binomial distribution

P(X = k) = \binom{n}{k} p^k(1-p)^{n-k}

binomial distribution

P(A \cap B) = P(A)P(B)

independent events

Symbols

$n$	number of trials
$k$	number of successes
$p$	probability of success

Example

Two sixes: $\dfrac{1}{6} \cdot \dfrac{1}{6} \approx 0.02778$ . Exactly two heads in five tosses: $\binom{5}{2}\left(\dfrac{1}{2}\right)^5 = 10 \cdot \dfrac{1}{32} = 0.3125$ .

The binomial coefficient counts the orders. Without it you only get the probability of one particular order.

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Part of Mathematics R1: Probability and combinatorics.