Question

Consider the experiment of tossing a coin. If the coin shows head, toss it again, but if it shows tail, then throw a die. Find the conditional probability of the event that 'the die shows a number greater than $4$', given that 'there is atleast one tail'.

Answer 1

The sample space for the given event is,

$\left\{HH,HT,T1,T2,T3,T4,T5,T6\right\}$

Let E be the event that dice show the number greater than 4 and f be the event of getting at least one tail.

$E=\left\{T5,T6\right\}$ and $F=\left\{HT,T1,T2,T3,T4,T5,T6\right\}$

$P\left(E\right)=P\left(T5\right)+P\left(T6\right)$

$=\frac{1}{12}+\frac{1}{12}$

$=\frac{1}{6}$

$P\left(F\right)=P\left(HT\right)+P\left(T1\right)+P\left(T2\right)+P\left(T3\right)+P\left(T4\right)+P\left(T5\right)+P\left(T6\right)$

$=\frac{1}{4}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}+\frac{1}{12}$

$=\frac{3}{4}$

$E\cap F=\left\{T5,T6\right\}$

$P\left(E\cap F\right)=\frac{1}{6}$

$P\left(E|F\right)=\frac{P\left(E\cap F\right)}{P\left(F\right)}$

$=\frac{\frac{1}{6}}{\frac{3}{4}}$

$=\frac{2}{9}$