Each of the persons A and B independently tosses three fair coins....

Each of the persons A and B independently tosses three fair coins.

Let $X$ be the number of heads obtained by $A$ and $Y$ be the number of heads obtained by $B$.

Both $X$ and $Y$ are binomial variate with parameters, $n = 3 \text{ and } p = \dfrac{1}{2}$

Probability that both $A$ and $B$ obtain the same number of heads

$= P\left(X = 0\right)P\left(Y = 0\right) + P\left(X = 1\right)P\left(Y = 1\right) + P\left(X = 2\right)P\left(Y = 2\right) + P\left(X = 3\right)P\left(Y = 3\right)$

$= {\left[C0 3 {\left(\dfrac{1}{2}\right)}^{3}\right]}^{2} + {\left[C1 3 {\left(\dfrac{1}{2}\right)}^{3}\right]}^{2} + {\left[C2 3 {\left(\dfrac{1}{2}\right)}^{3}\right]}^{2} + {\left[C3 3 {\left(\dfrac{1}{2}\right)}^{3}\right]}^{2}$

$= {\left[1 \times {\left(\dfrac{1}{2}\right)}^{3}\right]}^{2} + {\left[3 \times {\left(\dfrac{1}{2}\right)}^{3}\right]}^{2} + {\left[3 \times {\left(\dfrac{1}{2}\right)}^{3}\right]}^{2} + {\left[1 \times {\left(\dfrac{1}{2}\right)}^{3}\right]}^{2}$

$= {\left(\dfrac{1}{2}\right)}^{6}\left[1 + 9 + 9 + 1\right]$

$= \dfrac{20}{64}$

$= \dfrac{5}{16}$