Even vs Odd Heads and Coin Fairness

This is a three-part problem about the parity of heads in coin flips. **Part 1.** You flip a fair coin 10 times. What is the probability of getting an even number of heads? What about an odd number of heads? **Part 2.** You flip a single coin $n \geq 1$ times (where the coin has some unknown probability $p$ of landing heads). You observe that the probability of getting an even number of heads equals the probability of getting an odd number of heads. Does this imply the coin is fair (i.e., $p = 1/2$)? **Part 3.** Now you have 10 different coins with unknown head probabilities $p_1, p_2, \ldots, p_{10}$. You flip each coin once. You observe that the probability of the total number of heads being even equals the probability of it being odd. Does this imply that at least one of the coins is fair?