Question

Three fair coins are tossed simultaneously. If X denotes the number of heads, find the probability distribution of X.

Answer 1

Since three coins are tossed simultaneously.

Therefore sample space $S$ will be: $\{HHH,HHT,HTH,THH,HTT,THT,TTH,TTT\}$

Given that $X$ denotes the no. of heads such that

$X\left(0\right)=TTT=1$

$X\left(1\right)=HTT,THT,TTH=3$

$X\left(2\right)=HHT,HTH,THH=3$

$X\left(3\right)=HHH=1$

Total no. of outcomes $=2^n$

whereas,

$n=$ no. of coins tossed

$\therefore$ Total no. of outcomes $=2^3=8$

Therefore,

$P(X=0)=P\left(TTT\right)=\frac{1}{8}$

$P(X=1)=P(HTT,THT,TTH)=\frac{3}{8}$

$P(X=2)=P(HHT,HTH,THH)=\frac{3}{8}$

$P(X=3)=P\left(HHH\right)=\frac{1}{8}$

Therefore probability distribution of $X$-

$X$	$P\left(X\right)$
$0$	$\frac{1}{8}$
$1$	$\frac{3}{8}$
$2$	$\frac{3}{8}$
$3$	$\frac{1}{8}$