Problem of the Day #2

POTD November 21, 2023
Author: munch

Heather and Thomas play a game involving counters, in which a coin is flipped repeatedly. If the coin lands heads, then Thomas gives Heather a counter, and if the coin lands tails, then Heather gives Thomas a counter. Both Heather and Thomas start with two counters each, and the winner is the first person who acquires all four counters. If the coin is weighted so that the probability of heads is \frac{3}{5}, and the probability of tails is \frac{2}{5}, then the probability that Heather wins the game can be expressed in the form \frac{m}{n}, where m and n are relatively prime positive integers. Find m+n.