Chapter 11 Probability

Chapter 11: Probability 11.1 Basic Concepts 11.2 Events Involving “Not” and “Or” 11.3 Conditional Probability and Events Involving “And” 11.4 Binomial Probability 11.5 Expected Value and Simulation

Section 11-4 Binomial Probability

Binomial Probability Construct a simple binomial probability distribution. Apply the binomial probability formula for an experiment involving Bernoulli trials.

Binomial Probability Distribution The spinner below is spun twice and we are interested in the number of times a 2 is obtained (assume each sector is equally likely). Think of outcome 2 as a “success” and outcomes 1 and 3 as “failures.” The sample space is 2 1 3 S = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}.

Binomial Probability Distribution When the outcomes of an experiment are divided into just two categories, success and failure, the associated probabilities are called “binomial.” Repeated trials of the experiment, where the probability of success remains constant throughout all repetitions, are also known as Bernoulli trials.

Binomial Probability Distribution If x denotes the number of 2s occurring on each pair of spins, then x is an example of a random variable. In S, the number of 2s is 0 in four cases, 1 in four cases, and 2 in one case. Because the table on the next slide includes all the possible values of x and their probabilities, it is an example of a probability distribution. In this case, it is a binomial probability distribution.

Probability Distribution for the Number of 2s in Two Spins x P(x) 1 2

Binomial Probability Formula In general, let n = the number of repeated trials, p = the probability of success on any given trial, q = 1 – p = the probability of failure on any given trial, and x = the number of successes that occur. Note that p remains fixed throughout all n trials. This means that all trials are independent. In general, x successes can be assigned among n repeated trials in nCx different ways.

Binomial Probability Formula When n independent repeated trials occur, where p = probability of success and q = probability of failure with p and q (where q = 1 – p) remaining constant throughout all n trials, the probability of exactly x successes is given by

Example: Finding Probability in Coin Tossing Find the probability of obtaining exactly three heads in five tosses of a fair coin. Solution This is a binomial experiment with n = 5, p = 1/2, q = 1/2, and x = 3.

Example: Finding Probability in Dice Rolling Find the probability of obtaining exactly two 3’s in six rolls of a fair die. Solution This is a binomial experiment with n = 6, p = 1/6, q = 5/6, and x = 2.

Example: Finding Probability in Dice Rolling Find the probability of obtaining less than two 3’s in six rolls of a fair die. Solution We have n = 6, p = 1/6, q = 5/6, and x < 2.

Example: Finding the Probability of Hits in Baseball A baseball player has a well-established batting average of 0.250. In the next series he will bat 10 times. Find the probability that he will get more than two hits. Solution In this case n = 10, p = 0.250, q = 0.750, and x > 2.

Example: Finding the Probability of Hits in Baseball Solution (continued)