Question

A die is rolled. Consider events:

A = {1, 2, 5}, B = {3, 5}, C = {2, 3, 4, 5}

and hence find:

1. P(A|C) and P(C|A)
2. P(A ∩ B|C) and P(A ∪ B|C)

Answer 1

S = {1, 2, 3, 4, 5, 6}

A = {1, 2, 3}

n(A) = 3

B = {3, 5}

n(B) = 2

C = {2, 3, 4, 5}

n(C) = 4

i.

A ∩ C = {2, 5}

n(A ∩ C) = 2

P(A|C) = `(n(A ∩ C))/(n(C))`

= `2/4`

= `1/2`

P(C|A) = `(n(A ∩ C))/(n(A))`

= `2/3`

ii.

P(A ∩ B|C)

A ∩ B = {5}

(A ∩ B) ∩ C = {5}

n = ((A ∩ B) ∩ C) = 1

P(A ∩ B|C) = `(((A ∩ B) ∩ C))/(n(C))`

= `1/4`

P(A ∪ B|C)

A ∪ B = {1, 2, 3, 5}

(A ∪ B) ∩ C = {2, 3, 5}

n((A ∪ B) ∩ C) = 3

P(A ∪ B|C) = `(n((A ∪ B) ∩ C))/(n(C))`

= `3/4`