Question

Each of two persons $A$ and $B$ toss three fair coins. The probability that both get the same number of heads is

• A. $\frac{3}{8}$
• B. $\frac{1}{9}$
• C. $\frac{5}{16}$
• D. $\frac{7}{16}$

Answer 1

The correct option is C $\frac{5}{16}$

Let X be the number of heads obtained by A and Y be the number of heads
obtained by B. Note that both X and Y are binomial variate with
parameters $n=3$ and $p=1/2$.
Probability that both A and B obtain the same number of heads is
$=P(X=0)P(Y=0)+P(X=1)P(Y=1)+P(X=2)P(Y=2)+P(X=3)P(Y=3)$
$={\left[{}^3{C}_0{\left(\frac{1}{2}\right)}^3\right]}^2+{\left[{}^3{C}_1{\left(\frac{1}{2}\right)}^3\right]}^2+{\left[{}^3{C}_2{\left(\frac{1}{2}\right)}^3\right]}^2+{\left[{}^3{C}_3{\left(\frac{1}{2}\right)}^3\right]}^2$
$={\left(\frac{1}{2}\right)}^6[1+9+9+1]=\frac{20}{64}=\frac{5}{16}$.