Question 75
In a bag are $n$ fair, six-sided dice whose faces are colored white and red in such a way that the total numbers of white and red sides are equal. Let $p$ be the probability that the same color comes up twice when taking one die randomly out of the bag and throwing it twice. Let $q$ be the probability that the same color comes up twice when taking two dice randomly out of the bag and throwing them at the same time. Prove that
$$
p+(n-1) q=\frac{n}{2}
$$
$$
p+(n-1) q=\frac{n}{2}
$$