Slide 1Tree DiagramSlide 3Slide 4Slide 5Slide 6Slide 7Example 4 – A couple plans to have children until they have a girl or until they have three children. What is the probability they will have a girl among their children?Simulation:•Step 1: Probability Model–P(girl) = 0.49–P(boy) = 0.51More boys than girls are born. Boys have higher infant mortality so sexes even out.•Step 2: Assign digits–00, 01, 02, …, 48 = girl (49/100 = 0.49)–49, 50, 51, …, 99 = boy (51/100 = 0.51)•Step 3: Simulate repetitions•From table of random digits: 69 05 16 48 17 87 17 40 95 17 84 53 40 64 89 87 20 B G G G G B G G B G B B G B B B G•Estimated probability P(at least one girl) = 9 / 10 = 0.90•True probability is 0.867Tree Diagram•Tree Diagram–Useful tool for organizing more complicated probabilities–Graphical form R–Represents different stages of a probability model by different “branchings” –Multiplication up the branches denotes intersections of events (considering conditions)–Adding the resulting intersections’ probabilities can give us the probability of a union•Compute true probability this couple will have a girlBG0.510.49BG0.510.49BG0.510.491. P(G) = 0.492. P(BG) = (0.51)(0.49) = 0.2503. P(BBG) = (0.51)2(0.49) = 0.127P(G in three) = 0.49 + 0.250 + 0.127 = 0.867BG0.510.49BG0.510.49BG0.510.49What is the only way to NOT have a GIRL in three tries?BBBP(BBB) = (0.51)(0.51)(0.51) = 0.132Complement Rule: P(AC) = 1 – P(A)P(having a girl)= 1 – P(BBB) = 1 – 0.132 = 0.867Example 5 – Suppose a patient has been told he needs a kidney transplant. He wants to know probability of surviving at least five more years. He is told 90% survive the transplant operation. Of those surviving operation, 60% of transplants are successful and the other 40% need dialysis. Proportion surviving at least five years is 70% for those with new working and 50% for those returning to dialysis. SurviveSurgery0.900.10DieDialysis0.600.40New Kidney5-YearSurvival0.700.30Die5-YearSurvival0.500.50DieStep 1: Probability ModelExample 5 – Suppose a patient has been told he needs a kidney transplant. He wants to know probability of surviving at least five more years. He is told 90% survive the transplant operation. Of those surviving operation, 60% of transplants are successful and the other 40% need dialysis. Proportion surviving at least five years is 70% for those with new working and 50% for those returning to dialysis. Step 2: Assign digitsStage 1: Surgery survival–Survive: 1, 2, 3, 4, 5, 6,7, 8, 9 (9 of 10 unique digits)–Die: 0 (1 of 10 unique digits)Stage 2: Success / Dialysis–Succeeds: 0, 1, 2, 3, 4, 5 (6 of 10 unique digits)–Dialysis: 6, 7, 8, 9 (4 of 10 unique digits)Stage 3A: Survive 5 years with Kidney–Survive: 0, 1, 2, 3, 4, 5, 6 (7 of 10 unique digits)–Die: 7, 8, 9 (3 of 10 unique digits)Stage 3B: Survive 5 years with Dialysis–Survive: 0, 1, 2, 3, 4, (5 of 10 unique digits)–Die: 5, 6, 7, 8, 9 (5 of 10 unique digits)NOTE: Stage 3 depends on outcome of Stage 2 NOT independent.Example 5 – Suppose a patient has been told he needs a kidney transplant. He wants to know probability of surviving at least five more years. He is told 90% survive the transplant operation. Of those surviving operation, 60% of transplants are successful and the other 40% need dialysis. Proportion surviving at least five years is 70% for those with new working and 50% for those returning to dialysis. Step 3: Simulate repetitions Repetition 1 Repetition 2 Repetition 3 Repetition 4Stage 1 3 Survive 4 Survive 8 Survive 9 SurviveStage 2 8 Dialysis 8 Dialysis 7 Dialysis 1 KidneyStage 3 4 Survive 4 Survive 8 Die 8 Die•Patient survives 2 of our 4 repetitions (2 / 4 = 0.50)•Many repetitions are not something we would want to do by hand…•Using computer generated simulations, the “actual” probability of surviving for at least 5 years is
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