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

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 
$\therefore P(X=0)P(Y=0)+P(X=1)P(Y=1)+P(X=2)$$P(Y=2)+P(X=3)P(Y=3)$
                 $\begin{array}{l}= {\left[{ }^3{C}_0{\left(\frac{1}{2}\right)}^3\right]}^2+{\left[{ }^3{C}_1{\left(\frac{1}{2}\right)}^3\right]}^2\\ +{\left[{ }^3{C}_2{\left(\frac{1}{2}\right)}^3\right]}^2+{\left[{ }^3{C}_3{\left(\frac{1}{2}\right)}^3\right]}^2\\ = {\left(\frac{1}{2}\right)}^6[1+9+9+1]=\frac{20}{64}=\frac{5}{16}.\end{array}$