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’.

Answer 1

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