Question

A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed twice, find the probability distribution of number of tails.

Answer 1

Let X denotes the random variable which denotes the number of tails when a biased coin is tossed twice.

So, X may have value 0,1 or 2.

Since, the coin is biased in which head is 3 times as likely to occur as a tail.

$\therefore$ $P\left(H\right)=\frac{3}{4}andP\left(T\right)=\frac{1}{4}$ $P(X=0)=PHH={\left(\frac{3}{4}\right)}^2=\frac{9}{16}$

P(X=1)=P (one tail and one head)

=P{HT,TH}=P{HT}+P{TH}=P{H}P{T}+P{T}P{H}

$=\frac{3}{4}\times \frac{1}{4}+\frac{1}{4}\times \frac{3}{4}=\frac{3}{16}+\frac{3}{16}=\frac{6}{16}=\frac{3}{8}$

P(X=2)=P (two tails)= $PTT=PT.PT={\left(\frac{1}{4}\right)}^2=\frac{1}{16}$

Therefore, the required probability distribution is as follows

$\begin{array}{cccc}X& 0& 1& 2\\ P\left(X\right)& \frac{9}{16}& \frac{3}{8}& \frac{1}{16}\end{array}$