Question

How to calculate the P(A|B) of a 8 sided die that is rolled one time?

Event A occurs whenever a less than 2 is rolled. Event B occurs whenever an even number is rolled. Event C occurs whenever a 1 or 3 is rolled. What are the complements, probabilities, the intersections and the unions and are they statistically independent?

what are the P(A|B) and the P(C|A)?

Answer 1

A: Less than 2 is rolled

P(A) = 1/8

B: Even number is rolled that is the number rolled is from {2, 4, 6, 8}

P(B) = 4/8

C: number rolled is out of {1, 3}

P(C) = 2/8

The complements here are computed as:

P(A^c) = 1 - P(A) = 1 - (1/8) = 7/8

Therefore 7/8 is the required probability here.

P(B^c) = 1 - P(B) = 1 - (4/8) = 4/8

Therefore 4/8 = 1/2 is the required probability here.

P(C^c) = 1 - P(C) = 1 - (2/8) = 6/8

Therefore 6/8 = 3/4 is the required probability here.

b) The probabilities here are computed as:

P(A B) = P(A or B) = 5/8

P(A B) = P(A and B) = 0 as there is no intersection of events here.

P(A | B) = P(A and B) / P(B) = 0

Therefore 0 is the conditional required probability here.

P(A C) = P(A or C) = 2/8 = 1/4

P(A C) = P(A and C) = 1/8

P(C | A) = P(A and C) / P(A) = (1/8)/(1/8) = 1

Therefore 1 is the required conditional probability here.