Whenever the Lopsided Mint stamps a coin, its heads-bias $p$ is chosen uniformly at random between $0$ to $1$, which makes it have a probability $p$ of landing heads when flipped. Bob takes a random coin and flips it 10 times. Whats count(s) of heads is he most likely to obtain?
We're looking for the mode, not the expectation. The answer should not depend on $p$, since the coin's bias is unknown.
(You can solve this by doing some integrals, but there's an elegant argument with no calculation or algebra.)