Find the probability distribution of

number of tails in the simultaneous tosses of three coins

When three coins are tossed thrice, the sample space is

S= {HHH,HHT,HTH,HTT,THH,THT,TTH,TTT}

Which contains eight equally likely sample points.

Let X represent the number of tails.

Then, X can take values 0,1,2, and 3.

P(X=0) = P(no tail) =P({HHH})=$\frac{1}{8}$,

P(X=1) =P (one tail and two heads show up)

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

P(X=2) = P(two tails and one head show up)

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

and P(X=3) =P (three tails show up) =P({TTT}) =$\frac{1}{8}$.

Thus , the probability distribution is as follows

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