3.4 The Binomial Distribution

Binomial Trials A binomial experiment has the properties: The number of trials in the experiment is fixed. The only outcomes are “success” and “failure.” The probability of success in each trial is the same. The trials are independent of each other.

Probabilities in Bernoulli Trials In a binomial experiment in which the probability of success in any trial is p, the probability of exactly x successes in n independent trials is given by

Ex 1. A card is drawn from a standard 52-card deck Ex 1. A card is drawn from a standard 52-card deck. If drawing a club is considered a success, find the probability of exactly one club in 4 draws (with replacement). no clubs in 5 draws (with replacement). At least 3 clubs in 5 draws (with replacement).

Mean, Variance, and Standard Deviation of a Random Variable X If X is a binomial random variable associated with a binomial experiment consisting of n trials with probability of success p and probability of failure q, then the mean variance, and standard deviation of X are

Ex 2. A baseball player has a 0.241 batting average. In 40 at-bats, how many hits is he expected to make. Find the standard deviation.

Ex 3: If the probability that a certain tennis player will serve an ace is 1/4, what is the probability that she will serve exactly two aces out of five serves?

Ex 4: From experience, the manager at a book store knows that 40% of the people who are browsing the store will make a purchase. What is the probability that among ten people who are browsing the store, at least three will make a purchase?