At most craps tables in gambling houses, the house will give odds of 6 to 1 against rolling a 4 the “hard way,” that is, with a pair of 2’s as contrasted with a 3 and a 1, which is the “easy way.” A bet made on a “hard-way” 4 wins if a pair of 2’s show before either a 7 is rolled or a 4 is made the “easy way”; otherwise it loses. What is the expectation purchased by a $1 bet on a “hard-way” 4?