Properties of the Binomial Probability Distributions 1- The experiment consists of a sequence of n identical trials 2- Two outcomes (SUCCESS and FAILURE ) are possible on each trial 3- The probability of success, denoted by p, does not change from trial to trial. Consequently, the probability of failure, denoted by q and equals to 1-p, does not change from trial to trial 4- The trials are independent.

Example :4 Fitness Test The Heart Association claims that only 10% of adults over 30 can pass the minimum requirements of Fitness Test. Suppose four adults are randomly selected and each is given the fitness test. Use the formula for a binomial random variable to find the probability distribution of x, where x is the number of adults who pass the fitness test. Graph the distribution.

According to a research only 5% of the cigarette smokers enter into a treatment program to help them quit smoking. In a random sample of 200 smokers, let x be the number who enter into a treatment program. A-) Explain why x is a binomial r.v. B-) What is the value of p? Interpret this value. c-) What is the expected value of x? Interpret this value.

Example: Purchase Decision Consider the purchase decisions of the next three customers who enter the clothing store. On the basis of past experience, the store manager estimates the probability that any one customer will make a purchase is 0.30 Q: What is the probability that two of the next three customers will make a purchase?

Example-1: An insurance company sells a 10,000 TRL 1-year term insurance policy at an annual premium of 290 TRL. Based on many year’s information, the probability of death during the next year for a person of customer’s age, sex, health etc. is Q: What is the expected gain (amount of money made by the company) for a policy of this type?

Example: 1 Test the following function to determine whether it is a probability function. If it is not, try to make it into a probability function S(x) = (6 - |x – 7|) / 36, for x = 2, 3, 4, 5, 6, 7,..., 11, 12 a. List the distribution of probabilities and sketch a histogram. b. Do you recognize S(x)? If so, identify it.

Example: 2 The College Board website provides much information for students, parents, and professionals with respect to the many aspects involved in Advanced Placement (AP) courses and exams. One particular annual report provides the percent of students who obtain each of the possible AP grades (1 through 5). The 2008 grade distribution for all subjects was as follows: AP Grade Percent a. ) Express this distribution as a discrete probability distribution. b. ) Find the mean and standard deviation of the AP exam scores for 2008.

MATHEMATICAL EXPECTATION Q-1 : What is our mathematical expectation if we will receive 20 TL if and only if a coin comes up Head? Q-2 : What is our mathematical expectation if we buy 1 of 2000 raffle tickets issued for a first prize of a smart phone set worth 540 USD, a second prize of a bike set worth 180 USD a third prize of a pocket radio set worth 40 USD. Q-3 : To handle a liability suit, A lawyer has to decide whether to charge a straight fee of 2400 TL or a contingent fee of 9600 TL which she will get only if her client wins. How does she feel about her client’s chances if she prefers the straight fee of 2400 TL?

Q: A clothing manufacturer must decide whether to spend a considerable sum of money to build a new factory. The following table represent the information about the profits and deficits called Payoff Table: New Factory BUILTNew Factory NOT Built Good Sales Year 451,000220,000 Poor Sales Year- 110,000 22,000 If the clothing manufacturer feels that the probabilities for a good sales year or a poor sales year are, respectively, 0.40 and 0.60, would building the new factory maximize his expected profit?

Example :4 The Heart Association claims that only 10% of adults over 30 can pass the minimum requirements of Fitness Test. Suppose four adults are randomly selected and each is given the fitness test. Use the formula for a binomial random variable to find the probability distribution of x, where x is the number of adults who pass the fitness test. Graph the distribution.