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. The conditional probability of the event 'the coin shows a tail', given that 'atleast one die shows a $3$'.

• A. $\frac{1}{6}$
• B. $\frac{1}{2}$
• C. $1$
• D. $0$

Answer 1

The correct option is D $0$

The experiment is explained below in the tree diagram:

The sample space of the given experiment is given below
$S=\left\{\begin{array}{c}(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)\\ (6,1),(6,2),(6,3),(6,4),(6,5),(6,6)\\ 1H,2H,4H,5H,1T,2T,4T,5T\end{array}\right\}$
Let $E$ be the event that 'the coin shows a tail' and $F$ be the event that 'atleast one die shows a $3$'.
$\Rightarrow E=\left\{\begin{array}{c}1T,2T,4T,5T\end{array}\right\}$ and $F=\left\{\begin{array}{c}(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)(6,3)\end{array}\right\}$
Clearly, $E\cap F=\varphi$ $\Rightarrow P(E\cap F)=0$
Now, we know that by definition of conditional probability
$P\left(E/F\right)=\frac{P(E\cap F)}{P\left(F\right)}$
$\Rightarrow P\left(E/F\right)=\frac{0}{P\left(F\right)}=0$