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Question
A black and a red dice are rolled.
Find the conditional probability of obtaining a sum greater than 9, given that the black die resulted in a 5.
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Solution
Let x denote the outcome on the black die and y denote the outcome on the red die, then sample space is
S = {(x, y): x, y ∈ (1, 2, 3, 4, 5, 6)}, which contain 6 × 6 = 36 equally likely simple events.
E: 'sum greater than 9' and F: 'black die resulted in a 5'
E = {(6, 4), (4, 6), (5, 5), (5, 6), (6, 5), (6, 6)}
and F = {(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)}
⇒ E ∩ F = {(5, 5), (5, 6)}
`P (E) = 6/36, P(F) = 6/36, P (E cap F) = 2/36`
Required probability= P(E|F)
`(P(E cap F))/(P(F)) = (2/36)/(6/36)`
`= 2/6 = 1/3`
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