Question
A dealer at a casino holds a game that is played as follows: The player first pays $\$d$ to the dealer, where $d$ is some positive integer less than $100.$ The dealer then intends to flip a coin $d$ times. If at any point two consecutive heads are flipped, then the dealer stops flipping the coin, and the player wins and is awarded $\$100.$ Otherwise, the player loses (and is awarded nothing). A player pays the dealer $\$29.$ The dealer weights the coin before flipping it, changing the probability of flipping heads such that the player's expected net profit is non-positive. The player is not aware of this re-weighing. Find the maximum probability that the dealer should choose.