Finite Probabilty Help

Question

I need help figuring out a few problems.

 

1. For two events, M and N, P(M)=0.6, P(N|M)=0.3, and P(N|M')=0.3. Find P(M|N').

 

2. Suppose you have 3 jars with the following contents. Jar 1 has 1 white ball and 1 black ball. Jar 2 has 2 white balls and 1 black ball. Jar 3 has 1 white ball and 2 black balls. One jar is to be selected, and then 1 ball is to be drawn from the selected jar. The probabilities of selecting the first, second, and third jars are 1/3, 1/2, and 1/6respectively. Find the probability the ball was drawn from Jar 2, given that the ball is white.

Answer 1

1.   Find P(M|N').  Use a Venn diagram with circles M and N overlapping; probability in their intersection is b, and probability in circle M but outside circle N is a, and probability in circle N but outside circle M is c.

 

P(N|M) = P(N∩M) / P(M)

   .3     = P(N∩M) / .6  therefore P(N∩M) = 0.18 = b (and thus a = 0.42)

 

P(N|M') = P(N∩M') / P(M')

     .3    =   P(N∩M') / .4 therefore  P(N∩M') = .12 = c and thus P(N) = b+c = 0.3 so P(N') = 0.7

 

P(M|N') = P(M∩N') / P(N')

            = .42 / .7 = 0.6   Refer to Venn diagram to visualize  P(M∩N') = 0.42

 

Final Answer: 0.6