Binomial Distribution

Study guide: Exam P-1

The Binomial Distribution X \sim \text{Bin}(n, p) models the number of successes in n independent Bernoulli trials, each with probability of success p.

• Requires trials to be independent, each trial to have exactly two outcomes, and p to be constant across trials

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

k = 0, 1, \ldots, n

• E[X] = np and \text{Var}(X) = np(1-p)

Number of Claims in a Group Policy A group of 10 policyholders each independently file a claim with probability 0.3. Find the probability that exactly 4 file claims.

Answer X \sim \text{Bin}(10, 0.3), so: P(X = 4) = \binom{10}{4}(0.3)^4(0.7)^6 = 210 \times 0.0081 \times 0.117649 \approx 0.2001