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The Binomial Distribution Chapter 7 Section 3 When a probability problem can be reduced to two outcomes or only has two outcomes, it is called a binomial experiment. It must also meet the following requirements: a.) must be a fixed number of trials b.) outcomes of trials must be independent c.) probability for a success must remain the same for each trial. Binomial Experiment The outcomes of a binomial experiment and the corresponding outcomes are called a binomial distribution. Notation for Binomial Distribution: P(S) symbol for the probability of success P(F) symbol for the probability of failure p numerical probability of success q numerical probability of failure P(S) = p P(F) = 1 – p = q n number of trials X number of successes in n trials Binomial Distribution In a binomial experiment, the probability of exactly X successes in n trials is: n! X n X P(X) p q (n X)!X! Binomial Probability Formula A coin is tossed 3 times. Find the probability of getting exactly two heads. This is a binomial experiment because: 1.) Fixed number of trials (n = 3) 2.) There are only tow outcomes for each trial (heads or tails) 3.) The outcomes are independent 4.) The probability of success (heads) is ½ in each case. Example #1 We can solve this problem simply by looking at the sample space: Example #1 Cont. In this problem: n = 3, X = 2, p = ½ and q = ½ n! X n X P(X) p q (n X)!X! 2 3! 1 1 p(2 heads) Example #1 Cont. (3 2)!2! 2 2 (32) A survey found that one out of five Americans say he or she has visited a doctor in any given month. If 10 people are selected at random, find the probability that exactly 3 will have visited a doctor last month. n= X= Example #2 p= q= A survey from Teenage Research Unlimited (Northbrook, Illinois) found that 30% of teenage consumers receive their spending money from part-time jobs. If 5 teenagers are selected at random, find the probability that at least 3 of them have part time jobs. (To find the probability of at least, we must find the probability of 3, 4, and 5 and then add them together) Example #3 Public Opinion reported that 5% of Americans are afraid of being alone in a house at night. If a random sample of 20 Americans is selected, find the probability there are exactly 5 people in the sample who are afraid of being alone at night. Example #4 Twenty-five percent of the customers entering a grocery store between 5 PM and PM use an express checkout. Consider 5 randomly selected customers and let x denote the number among the five who use the express checkout. What is p(2)? What is (P<=1) What is P(>=2) Example #5 Mean: n p Variance: n pq 2 Standard Deviation: 2 Mean, Variance, and Standard Deviation for the Binomial Distribution A coin is tossed 4 times. Find the mean, variance, and standard deviation. A die is rolled 480 times. Find the mean, variance and standard deviation. Example #6 and #7 Geometric Distribution Chapter 7 Section 5 A geometric random variable is defined as the number of trials (x) until the first success is observed. The probability distribution of x is called the geometric probability distribution. Geometric Random Variable p(x) q x1 p Geometric Probability Distribution Equation P(x n) 1 q n And P(x n) 1 (1 q ) n Cumulative Geometric Distribution Equation Expected value 1 E(x) p You have left your lights on and your car battery has died. Assume that 20% of students who drive to school carry jumper cables. What is the probability that the first student you ask has jumper cables? What is the probability that 3 or fewer students must be stopped? What is the expected number before someone will have jumper cables? Example #1 One out of seven scratch off lottery tickets has a prize. You keep buying tickets until you win a prize, then you stop. a. b. c. d. Find the probability that you buy 4 tickets. Find the probability that you buy 3 or fewer tickets Find the probability that you buy more than four tickets What is the expected number of tickets bought? Example #2