# Bayes' Theorem?

1. Three professors administer a common exam to their classes of 40 students each.Suppose that 32 of the students in class A,30 of the students in class B and so 30 of the students in class C pass the exam.If a student who has passed the exam is chosen at random,what is the propability that the student is in class A?

2.The experiment is to choose at random either an urn or a box ,each containing some yellow and green marbles,and then select a marble .The urn has a 4 yellow and 6 green marbles.The bix has 7 yellow and 5 green marbles .Find the probability of each event.

a.) The urn was chosen given that the marble is yellow

b.) The box was chosen given that the marble is green.

Relevance

Bayes theorem says that P(A|B) = P(B|A) P(A)/P(B)

P(Being in A) = P(Being in B) = P(Being in C) = 40/120 =1/3

P(Passing) = (32+30+30)/120 = 92/120 = 23/30

P(Passing| Being in A) = 32/40 = 4/5

P(Passing| Being in B) = 30/40 = 3/4

P(Passing| Being in C) = 30/40 = 3/4

P(Being in A | Passing) = P(Passing | Being in A) * P(Being in A) / P(Passing) = ( 4/5 * 1/3 ) / 23/30 = 120/345 = 8/23

2. Same approach:

P(Y) =

P(G) =

P(Urn) =

P(Box) =

P(Y | Urn) =

P(G | Urn) =

P(Y | Box) =

P(G | Box) =

Solve:

P(Urn | Y) =

P(Box | G) =

I'll let you do it :-)