gen-define-the-binomial-distribution-and-give-one-short-worked-e-d1d8

Define the binomial distribution and give one short worked example.

# Binomial Distribution ## Definition The **binomial distribution** models the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes (success/failure) and the probability of success remains constant. **Parameters:** - n = number of trials - p = probability of success on each trial - X = number of successes **Probability mass function:** $$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$$ where $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ ## Worked Example **Problem:** A fair coin is flipped 5 times. What's the probability of getting exactly 3 heads? **Solution:** - n = 5 (number of flips) - p = 0.5 (probability of heads) - k = 3 (desired number of heads) $$P(X = 3) = \binom{5}{3} (0.5)^3 (0.5)^{2}$$ $$= \frac{5!}{3!2!} \times 0.125 \times 0.25$$ $$= 10 \times 0.03125$$ $$= 0.3125$$ **Answer:** The probability of getting exactly 3 heads is **0.3125 or 31.25%**.