Question

Use Bayes' theorem or a tree diagram to calculate the indicated probability. Round your answer to four decimal places.

Y₁, Y₂, Y₃ form a partition of S. P(X | Y₁) = .8, P(X | Y₂) = .4, P(X | Y₃) = .7, P(Y₁) = .4, P(Y₂) = .1. Find P(Y₁ | X).

P(Y₁ | X) =

According to a study in a medical journal, 202 of a sample of 5,990 middle-aged men had developed diabetes. It also found that men who were very active (burning about 3,500 calories daily) were a fourth as likely to develop diabetes compared with men who were sedentary. Assume that one-half of all middle-aged men are very active, and the rest are classified as sedentary. What is the probability that a middle-aged man with diabetes is very active? (Round your answer to four decimal places.)

Answer 1

using Bayes' theorem

P(Y1 I X) = P(X I Y1).P(Y1)/ P(X)

P(X) = P(X I Y1).P(Y1) + P(X I Y2).P(Y2)+P(X I Y3).P(Y3) = 0.8*0.4 + 0.4*0.1 +0.7*0.5 = 0.71

P(Y1 I X) = 0.8*0.4/0.71 = 0.4507

Therefore ,

P(Y1 I X)= 0.4507

note : P(Y3) = 1-P(Y1)-P(Y2) =0.5

P(active) =0.5

P(sedentary)= 0.5

P(diabetes) = 202/5990= 0.0337

P(diabetes)= P(diabetes I active ) .P(active)+ P(diabetes I sedentary).P(sedentary)

= P(diabetes I active ) .P(active) + 4.P(diabetes I active ).P(sedentary)

0.0337 = P(diabetes I active ).0.5 + 4.P(diabetes I active ).0.5

P(diabetes I active ) = 0.01348

To find

P(active I diabetes) = P(diabetes I active ).P(active) / P(diabetes)

= 0.01348*0.5 / 0.0337 = 0.2

Therefore P(active I diabetes)= 0.2