Question:
Coin $A$ is flipped three times and coin $B$ is flipped four times. What is the probability that the number of heads obtained from flipping the two fair coins is the same?

Answer:
The result will occur when both $A$ and $B$ have either $0,$ $1,$ $2,$ or $3$ heads, and these probabilities are shown in the table. \[
\begin{array}{ccccc}
\text{Heads} & 0 & 1 & 2 & 3 \\
\hline
{} & & & & \\[-9pt]
A & \dfrac{1}{8} & \dfrac{3}{8} & \dfrac{3}{8} & \dfrac{1}{8} \\[8pt]
\hline
{} & & & & \\[-9pt]
B & \dfrac{1}{16}& \dfrac{4}{16}& \dfrac{6}{16}& \dfrac{4}{16}
\end{array}
\] The probability of both coins having the same number of heads is \[
\frac{1}{8}\cdot \frac{1}{16} + \frac{3}{8}\cdot \frac{4}{16} + \frac{3}{8}\cdot \frac{6}{16} + \frac{1}{8}\cdot \frac{4}{16} = \boxed{\frac{35}{128}}.
\]