Chapter 8 Counting Principles: Further Probability Topics Section 8.4 Binomial Probability

Many probability problems are concerned with experiments in which an event is repeated many times. Many probability problems are concerned with experiments in which an event is repeated many times. Probability problems of this kind are called Bernoulli trials, or Bernoulli processes. Probability problems of this kind are called Bernoulli trials, or Bernoulli processes. In each case, some outcome is designated a success, and any other outcome is considered a failure. In each case, some outcome is designated a success, and any other outcome is considered a failure. Bernoulli trials problems are sometimes referred to as binomial experiments. Bernoulli trials problems are sometimes referred to as binomial experiments.

Binomial Experiments The following criteria must be met: 1.) The same experiment is repeated several times. 2.) There are only two possible outcomes, success and failure. 3.) The repeated trials are independent, so that the probability of success remains the same for each trial.

A single die is rolled four times in a row. Getting a 5 is a “success”, while getting any other number is a “failure”. A single die is rolled four times in a row. Getting a 5 is a “success”, while getting any other number is a “failure”. a.) Why is this a binomial experiment? Already classified outcomes as successes or failures. Already classified outcomes as successes or failures. Experiment is repeated several times. Experiment is repeated several times. Outcomes are independent when rolling a single die. Outcomes are independent when rolling a single die.

A single die is rolled four times in a row. Getting a 5 is a “success”, while getting any other number is a “failure”. A single die is rolled four times in a row. Getting a 5 is a “success”, while getting any other number is a “failure”. b.) Find the probability of having 3 successes followed by 1 failure. P(S  S  S  F) =

A single die is rolled four times in a row. Getting a 5 is a “success”, while getting any other number is a “failure”. A single die is rolled four times in a row. Getting a 5 is a “success”, while getting any other number is a “failure”. c.) Find the probability of having 3 successes and 1 failure, in any order. and 1 failure, in any order. Possible Outcomes SSSFSSFS SFSS FSSS Thus, P(3S  1F) = 4 ( ) = =

Binomial Probability Formula If p is the probability of success in a single trial of a binomial experiment, the probability of x successes and n-x failures in n independent repeated trials of the experiment is If p is the probability of success in a single trial of a binomial experiment, the probability of x successes and n-x failures in n independent repeated trials of the experiment is n = # of trials x = # of successes p = probability of success

Example: A single die is rolled 10 times. Find the probability of rolling exactly 7 fives. Example: A single die is rolled 10 times. Find the probability of rolling exactly 7 fives. n = # trials = 10 success: rolling a five x = # successes = 7 p = prob. of success = 1/6

Example: A restaurant manager estimates the probability that a newly hired waiter will still be working at the restaurant six months later is only 60%. For the five new waiters just hired, what is the probability that at least four of them will still be working at the restaurant in six months? Assume that the waiters decide independently of each other. Example: A restaurant manager estimates the probability that a newly hired waiter will still be working at the restaurant six months later is only 60%. For the five new waiters just hired, what is the probability that at least four of them will still be working at the restaurant in six months? Assume that the waiters decide independently of each other. n = 5 x = 4 or 5 p = 0.6

Example: A flu vaccine has a probability of 80% of preventing a person who is inoculated from getting the flu. A county health office inoculates 87 people. Find the probabilities of the following. Example: A flu vaccine has a probability of 80% of preventing a person who is inoculated from getting the flu. A county health office inoculates 87 people. Find the probabilities of the following. a.) Exactly 15 of the people inoculated get the flu. b.) No more than 3 of the people inoculated get the flu. c.) None of the people inoculated get the flu.