MATH Homework #3 Solutions. a) P(E F) b) P(E'UF') c) P(E' F') d) P(E F) a) P(E F) = P(E) * P(F E) P(E F) =.55(.20) =.11

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1 p Suppose P(E).55, P(F).40 and P(F E).20. Find a) P(E F) b) P(E'UF') c) P(E' F') d) P(E F) a) P(E F) P(E) * P(F E) P(E F).55(.20). b) P(E' F') P((E F)') P(E' F') P(E F) P(E' F')..89 c) P(E' F') P((E F)') P(E' F') P(E F) P(E' F') (P(E) + P(F) P(E F)) P(E' F') ( ) P(E' F').84.6 P(E F) d) P(E F) P(F) P(E F)

2 p Suppose P(E).50, P(F).20 and P(F E).30. Find a) P(E F) b) P(E F) c) P(E' F') d) P(E F) a) P(E F) P(E) * P(F E) P(E F).50(.30).5 b) P(E F) P(E) + P(F) P(E F) P(E F) c) P(E' F') P((E F)') P(E' F') P(E F) P(E' F') P(E F) d) P(E F) P(F) P(E F)

3 p Roll a fair die four times. What is the probability of at least one ace? Why is the answer not (/6) + (/6) + (/6) + (/6) (4/6) (2/3)? P( ace) P(No Aces) P( ace) ( 5 6 ) You do not use addition because the 4 rolls of the dice are not disjoint.

4 p Consider a two-stage experiment, where the event E is in the first stage and F is in the second stage, and the following probabilities. P(E') 3, P(F E'), P(E F) 4 Make a tree diagram including all branch probabilities and all terminal probabilties. P(E') 3 P(F E') 4 P(E F) P(E) P(E') P(F E) P(E F) P(E) P(F' E) P(F E) P(F' E') P(F E') P(F E) 8 P(E)*P(F E) 2 3 ( 8 ) P(E) 2 3 P(F' E) 7 8 P(E)*P(F' E) 2 3 ( 7 8 ) 7 P(E') 3 P(F E') 4 P(E') * P(F E') 3 ( 4 ) P(F' E') 3 4 P(E') * P(F' E') 3 ( 3 4 ) 4

5 p The American Cancer Society as well as the medical profession recommend that people have themselves checked annually for any cancerous growths. If a person has cancer, then the probability is 0.99 that it will be detected by a test. Furthermore, the probability that the test results will be positive when no cancer actually exits is 0.0. Government records indicate that 8% of the population in the vicinity of a paint manufacturing plant has some form of cancer. Find PV+ and PV-. P(Disease).08 P(No Disease) P(Disease) P(Positive Disease).99 P(Negative Disease) P(Positive Disease).99.0 P(Positive No Disease).0 P(Negative No Disease) P(Positive No Disease)..9 P(Positive Disease) P(Disease) * P(Positive Disease).08(.99).0792 P(Negative Disease) P(Disease) * P(Negative Disease).08(.0).0008 P(Positive No Disease) P(No Disease) * P(Positive No Disease).92(.0).092 P(Negative No Disease) P(No Disease) * P(Negative No Disease).92(.90).828 P(Positive) P(Positive Disease) + P(Positive No Disease) P(Negative) P(Negative Disease) + P(Negative No Disease) PV + P(Disease Positive) P(Disease Positive) P(Positive) PV P(No Disease Negative) P(No Disease Negative) P(Negative)

6 p The prevalence of breast cancer in the population of women between 40 and 50 years of age is (/). Assume the same sensitivity and specificity of mammography as in Example 3.34 above. a) Find PV+. b) Compare the risk of breast cancer before the test with the risk after a positive mammogram. c) Compare the risk of breast cancer before the test with the risk after a negative mammogram. P(Cancer) P(No Cancer) P(Cancer) 62 P(Positive Cancer).80 P(Negative Cancer) P(Positive Cancer) P(Negative No Cancer).90 P(Positive No Cancer) P(Negative No Cancer).90.0 P(Positive Cancer) P(Cancer) * P(Positive Cancer) (.80) P(Negative Cancer) P(Cancer) * P(Negative Cancer) (.20) P(Positive No Cancer) P(No Cancer) * P(Positive No Cancer) 62 (.0) P(Negative No Cancer) P(No Cancer) * P(Negative No Cancer) 62 (.90) P(Positive) P(Positive Cancer) + P(Positive No Cancer) P(Negative) P(Negative Cancer) + P(Negative No Cancer) 62 (.80) + (.0). 62 (.20) + (.90) a) PV + P(Cancer Positive) P(Cancer Positive) P(Positive) (.80) 62 (.80) + (.0)

7 b) P(Cancer) PV PV c) P(Cancer) P(Cancer Negative) P(Cancer Negative) P(Negative) (.20) (.20) + (.90)

8 p You go to a beach party. Two of you are bringing coolers with sandwiches. Your cooler contains ham sandwiches and 5 cheese sandwiches. Your friend's cooler has 6 ham and 9 cheese sandwiches. At the beach, someone takes a sandwich at random from one of the coolers/ It turns out to be a ham sandwich. What is the probability that the sandwich came from your cooler? First, sketch a tree diagram. Then use Baye's Method to complete the probability. YC Your Cooler OC Other Cooler H Ham C Cheese P(H YC) 6 P(H YC) 2 ( 6 ) P(YC) 2 P(C YC) 5 6 P(C YC) 2 (5 6 ) 5 P(H OC) 6 5 P(H OC) 2 ( 6 5 ) P(OC) 2 P(C OC) 9 5 P(C OC) 2 ( 9 5 ) P(YC H) P(YC H) P(H)

9 p A blood test to screen for a certain disease is not completely reliable, and medical officials are unsure as to whether the test should be routinely given. Suppose that 99.5% of those with the disease will show positive on the test, but that.2% of those who are free of the disease also show positive on the test. If 0.% of the population actually has the disease, find PV+ and PV-. D Disease, ND No Disease, P Positive, N Negative P(D).00 P(ND) P(D) P(P D).995 P(N D) P(P D) P(P ND).002 P(N ND) P(P ND) P(P D) P(D) * P(P D).00(.995) P(N D) P(D) * P(N D).00(.005) P(P ND) P(ND) * P(P ND).999(.002) P(N ND) P(ND) * P(N ND).999(.998) P(P) P(N) PV + P(D P) P(D P) P(P) PV P(ND N) P(ND N) P(N)