Problem 6

Bart Cassidy owns a bar in the Wild West, and wants to increase his revenue by introducing a gambling game for his customers. In return for 2 dollars, the customer rolls a 6-sided die and wins some amount of money depending on the roll. One of the frequent gamblers, Jesse Jones, observed that after three consecutive games, the probability that he ends up with more money than he started with is 55/108. What is the largest possible probability that two consecutive games bring in money? Give your answer as a decimal rounded to 5 decimal places.