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 con- trasted 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?

Question
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 con-
trasted 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?
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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 con- trasted 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?

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