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Question
Consider the experiment of throwing a die, if a multiple of 3 comes up, throw the die again and if any other number comes, toss a coin. Find the conditional probability of the event ‘the coin shows a tail’, given that ‘at least one die shows a 3’.
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Solution
The results of a given test are represented by a set.
Hence, the sample space in the test is S = {(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (6 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6), (1, H), (1, T), (2, H), (2, T), (4, H), (4, T), (5, H), (5, T)}
∴ n(S) = 20
Let E be the event that a coin shows tails and F be the event that the number 3 shows up on at least one dice.
E = {(1, T), (2, T), (4, T), (5, T)]
⇒ n (E) = 4
F = [(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (6, 3)]
∴ n(F) = 7
E ∩ F = 0 as there is no common point.
P(E) = `("Number of event occurrences")/("Total types") = (n(E))/(n(S)) = 4/20 = 1/5`
and P(F) = `(n(F))/(n(S)) = 7/20`
P(E ∩ F) = `(n(E ∩ F))/(n(S)) = 0/20 = 0`
∴ Required probability = `P(E/F) = (P(E ∩ F))/(P(F))`
`= 0/(7/20)`
= 0
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