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 ‘.
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 ‘.

Let E be the event that ‘the coin shows a tail’ and F be the event that ‘at least one die shows a 3 ‘.

\Rightarrow \mathrm{E}={1 \mathrm{~T}, 2 \mathrm{~T}, 4 \mathrm{~T}, 5 \mathrm{~T}} and \mathrm{F}={(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(6,3)}

\Rightarrow E \cap F=\varphi \Rightarrow P(E \cap F)=0 \ldots \ldots \ldots (i)

Now, we know that by definition of conditional probability, P(E \mid F)=\frac{P(E \cap F)}{P(F)}

\Rightarrow \mathrm{P}(\mathrm{E} \mid \mathrm{F})=\frac{0}{\mathrm{P}(\mathrm{F})}=0

\Rightarrow \mathrm{P}(\mathrm{E} \mid \mathrm{F})=0