6. In an election, there are two candidates, $A$ and $B$, who each have 5 supporters. Each supporter, independent of other supporters, has a $\frac{1}{2}$ probability of voting for his or her candidate and a $\frac{1}{2}$ probability of being lazy and not voting. What is the probability of a tie (which includes the case in which no one votes)?
Solution:
$$
63 / 256
$$

The probability that exactly $k$ supporters of $A$ vote and exactly $k$ supporters of $B$ vote is $\binom{5}{k}^{2} \cdot \frac{1}{2^{10}}$. Summing over $k$ from 0 to 5 gives
$$
\left(\frac{1}{2^{10}}\right)(1+25+100+100+25+1)=\frac{252}{1024}=\frac{63}{256} .
$$