A coin is tossed until a head appears or it has been tossed thrice. Given that head doesn't appear on…

A coin is tossed until a head appears or it has been tossed thrice.

Given that head doesn't appear on the first toss.

Total possible cases or sample space $=\left\{H,TH,TTH,TTT\right\}$

Probability of head $=P\left(H\right)=\frac{1}{2}$

Probability of first tail and second head$=P\left(TH\right)=\frac{1}{2}\times \frac{1}{2}=\frac{1}{4}$

Probability of first two tails and third head $=P\left(TTH\right)=\frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}=\frac{1}{8}$

Probability of three tails $=P\left(TTT\right)$$=\frac{1}{8}$

Let, $A=$Head doesn't appear on the first toss

$P\left(A\right)=P\left(T\right)=\frac{1}{2}$

$B=$Coin tossed thrice

$P(A\cap B)=P\left(TTH\right)+P\left(TTT\right)=\frac{1}{4}$

Probability of coin tossed three times, when first toss got not head  $P\left(\frac{B}{A}\right)=\frac{P(A\cap B)}{P\left(A\right)}$

$=\frac{{\displaystyle \frac{1}{4}}}{{\displaystyle \frac{1}{2}}}=\frac{1}{2}$