The problem I am thinking of is to find the expected number of tosses needed to get either 3 heads or 3 tails. It needs not be consecutive, e.g., Head->Head->Tail->Head is a possibility.
For a fair or biased coin, this is bounded by 3 moves minimum and 5 moves maximum. For a fair coin, the expected number of tosses is 4.125.
The solution states "A bias towards either heads or tails is a bias in favor of MORE OF THE SAME. Given that the stopping rule is 3 of the same, the expected number of tosses to end the game must decrease." I verified this statement by deriving the equation for a biased coin and indeed saw that the expected number of tosses peaks for a fair coin and falls when biased in either direction.
But I would like to get a better intuition of this without the help of equations. I only understand the solution in part, in that yes, the bias will favor of more of the same for heads if biased towards heads, but it also comes at the cost of less of the same for tails. So the solution statement seems to imply that more of the same for heads outweighs the less of the same for tails, hence reducing the expected value, but how does one determine this intutively without equations?