Consider the experiment of throwing a dice, if a multiple of 3 comes up, throw the dice 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 dice shows a 3’.

.: The sample space $S$ is given by

$S = \left\{ {\begin{array}{llllllllllllllllllll}{(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)}\\{(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}\\{(1,H),(1,T),(2,H),(2,T),(4,H)(4,T)}\\{(5,H),(5,T)}\end{array}} \right\}$

Solution

.: Let $E:$ 'the coin shows a tail' and $F:$

'atleast one dice shows up a 3 ',
$E = \{ (1,T),(2,T),(4,T),(5,T)\}$

and
$F = \{ (3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(6,3)\}$

$\Rightarrow E \cap F = (\phi ) = ,P(E) = 4/20,P(F) = 7/20,P(E \cap F) = 0/20$

Hence, the required probability

$= P(E|F) = \cfrac{{P(E \cap F)}}{{P(F)}} = \cfrac{0}{{P(F)}} = 0$

$(E \cap F{\rm{ is impossible event }})$