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• Example: Consider a recent study conducted by the personnel manager of a major computer software company. The study showed that 30% of employees who left the firm within two years did so primarily because they were dissatisfied with their salary, 20% left because they were dissatisfied with their work assignments, 12% of the former employees indicated dissatisfaction with both their salary and their work assignments. • Question: What is the probability that an employee who leaves within two years does so because of dissatisfaction with salary, dissatisfaction with work assignment or both? Assigning Probabilities Basic Requirements for Assigning Probabilities 2. The sum of the probabilities for all experimental outcomes must equal 1. P(E1) + P(E2) + . . . + P(En) = 1 where: n is the number of experimental outcomes Multiplication Law The multiplication law provides a way to compute the probability of the intersection of two events. The law is written as: P(A B) = P(B)P(A|B) Mutual Exclusiveness and Independence Do not confuse the notion of mutually exclusive events with that of independent events. Two events with nonzero probabilities cannot be both mutually exclusive and independent. If one mutually exclusive event is known to occur, the other cannot occur.; thus, the probability of the other event occurring is reduced to zero (and they are therefore dependent). Two events that are not mutually exclusive, might or might not be independent. The Sales of Automobiles for 300 days 0 automobile sold 54 days 1 automobile sold 117 days 2 automobile sold 72 days 3 automobile sold 42 days 4 automobile sold 12 days 5 automobile sold 3 days Total: 300 days Random Variables A random variable is a numerical description of the outcome of an experiment. A discrete random variable may assume either a finite number of values or an infinite sequence of values. A continuous random variable may assume any numerical value in an interval or collection of intervals. Discrete Probability Distributions The probability distribution is defined by a probability function, denoted by f(x), that provides the probability for each value of the random variable. The required conditions for a discrete probability function are: f(x) > 0 f(x) = 1 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 0.001 Q: What is the expected gain (amount of money made by the company) for a policy of this type? 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 1 20.9 2 21.3 3 24.1 4 19.4 5 14.3 a. ) Express this distribution as a discrete probability distribution. b. ) Find the mean and standard deviation of the AP exam scores for 2008. 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 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. 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. 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? Properties of Normal Distributions 1- The entire family of normal distribution is differentiated by its mean µ and its standard deviation σ. 2- The highest point on the normal curve is at the mean which is also the median and the mode of the distribution. 3- The mean of the distribution can be any numerical value: negative, zero or positive. 4- The normal distribution is symmetric 5- The standard deviation determines how flat and wide the curve is 6- Probabilities for the random variables are given by areas under the curve. The total area under the curve for the normal distribution is 1 7- Because the distribution is symmetric, the area under the curve to the left of the mean is 0.50 and the area under the curve to the right of the mean is 0.50 8- The percentage of values in some commonly used intervals are; a-) 68.3% of the values of a normal r.v are within plus or minus one st.dev. of its mean b-) 95.4% of the values of a normal r.v are within plus or minus two st.dev. of its mean c-) 99.7% of the values of a normal r.v are within plus or minus three st.dev. of its mean Find the area under the standard normal curve that lies 1- to the right of z = - 0.55 2- to the left of z= 0.84 3- to the right of z = 1.69 4- to the left of z = - 0.74 5- between z = 0.90 and z= 1.33 6- between z = -0.29 and z= 0.59 A sociologist has been studying the criminal justice system in a large city. Among other things, she has found that over the last 5 years the length of time an arrested person must wait between their arrest and their trial is a normally distributed variable x with µ=210 days and σ=20 days. Q-) What percent of these people had their trial between 160 days and 190 days after their arrest? Q-1 - MOBILE PHONE Assume that the length of time, x, between charges of mobile phone is normally distributed with a mean of 10 hours and a standard deviation of 1.5 hours. Find the probability that the mobile phone will last between 8 and 12 hours between charges. Q-2 ALKALINITY LEVEL The alkalinity level of water specimens collected from the River in a country has a mean of 50 milligrams per liter and a standard deviation of 3.2 milligrams per liter. Assume the distribution of alkalinity levels is approximately normal and find the probability that a water specimen collected from the river has an alkalinity level A-) exceeding 45 milligrams per liter. B-) below 55 milligrams per liter. C-) between 51 and 52 milligrams per liter. The grades of 400 students in a statistics course are normally distributed with mean µ = 65 and variance σ2=100 Q: Find the probability that a student selected randomly from this group would score within any interval given below. 1- A grade between 60 and 65 2- A grade between 70 and 65 3- A grade between 52 and 68 4- A grade that is greater than 85 5- A grade that is less than 72 6- A grade between 70 and 78 Example : 1 A normal distribution has the mean 74.4 Find its standard deviation if 10% of the area under the curve lies to the right of 100 Example : 2 A random variable has a normal distribution with standard deviation 10 . Find its mean if the probability is 0.8264 that it will take on a value less than 77.5 Example :3 For a certain random variable having the normal distribution, the probability is 0.33 that it will take on a value less than 245 and the probability is 0.48 that it will take on a value greater than 260. Find the mean and standard deviation of the random variable