5. Two people toss a coin: one tossed it 10 times, and the other 11 times. Find the probability that the second person gets more heads than the first.
Solution 1. Let $\mathcal{A}$ be the event that the second participant gets heads more times than the first participant.

Let $\mathcal{B}$ be the event that the second participant gets tails more times than the first participant. It is clear that these events are equally probable and do not intersect. Indeed, if the second participant got both heads and tails more times than the first, the number of his tosses should have been at least two more. On the other hand, these events are mutually complementary, so the sum of their probabilities is 1. Therefore, the probability of each of them is $\frac{1}{2}$.