z To determine the probability of 2 white shirts being selected we use formula: P(AB) = P(A) P(B|A) P ( W 1 and W 2 ) = P ( W 1 ) P ( W 2 | W 1 ) = (9/12)(8/11) = 0.55 General Multiplication Rule - Example A golfer has 12 golf shirts in his closet. Suppose 9 of these shirts are white and the others blue. He gets dressed in the dark, so he just grabs a shirt and puts it on. He plays golf two days in a row and does not do laundry. What is the likelihood both shirts selected are white? 157 Exercises 162:28 z 3 Defective and 17 Good Toothbrushes shipped z (a) Probability 1st 2 are Defective? z P(Defective #1) * P(Defective #2/Defective #1) z 3/20 * 2/19 = 6/380 = 0.01579 z (b) Probability NONE of 1st 2 are Defective? z P(Good #1) * P(Good #2/Good #1) z 17/20 * 16/19 = 272/380 = 0.7158
Exercises 163:32 z 3 Strangers, questions about their birthdays z (a) Probability ALL born Wednesday? z 1/7 * 1/7 * 1/7 = 0.002915 z (a) Probability NONE born Saturday? z 6/7 * 6/7 * 6/7 = 0.6297 z (c) Probability born on DIFFERENT days? z 7/7 * 6/7 * 5/7 = 0.6122 Contingency Tables A CONTINGENCY TABLE is a table used to classify sample observations according to two or more identifiable characteristics E.g. A survey of 150 adults classified each as to gender and the number of movies attended last month. Each respondent is classified according to two criteria—the number of movies attended and gender. 158
Contingency Tables - Example A sample of executives were surveyed about their loyalty to their company. One of the questions was, “If you were given an offer by another company equal to or slightly better than your present position, would you remain with the company or take the other position?” The responses of the 200 executives in the survey were cross-classified with their length of service with the company. What is the probability of randomly selecting an executive who is loyal to the company (would remain) and who has more than 10 years of service? 159 Event A 1 happens if a randomly selected executive will remain with the company despite an equal or slightly better offer from another company. Since there are 120 executives out of the 200 in the survey who would remain with the company P ( A 1 ) = 120/200, or .60. Event B 4 happens if a randomly selected executive has more than 10 years of service with the company. Thus, P(B 4 | A 1 ) is the conditional probability that an executive with more than 10 years of service would remain with the company. Of the 120 executives who would remain 75 have more than 10 years of service, so P(B 4 | A 1 ) = 75/120 . Contingency Tables - Example 159
Tree Diagrams A tree diagram is useful for portraying conditional and joint probabilities. It is particularly useful for analyzing business decisions involving several stages. A tree diagram is a graph that is helpful in organizing calculations that involve several stages. Each segment in the tree is one stage of the problem. The branches of a tree diagram are weighted by probabilities.