Question

15.1. Hardway, place, buy, and lay bets. (a) There are four hardway bets. An even dice total is said to be hard if both dice show the same number. Otherwise the total is said to be easy. For j 2, 3,4,5, a hard-2j wager is a bet that a hard 2j will appear before the first of an easy 2j and 7. Payoff odds are 7 to 1 on the hard 4 and hard 10, 9 to 1 on the hard 6 and hard 8. Find the house advantages.

the correct answer is

hardways: 4 or 10, 0.111111; 6 or 8, 0.090909.

Answer 1

There are 6 ways to roll a 7: 1-6, 2-5, 3-4, 4-3, 5-2, 6-1.
There are 2 ways to roll an easy 4: 1-3, 3-1
or an easy 10: 4-6, 6-4.
There are 4 ways to roll an easy 6: 1-5, 2-4, 4-2, 5-1
or an easy 8: 2-6, 3-5, 5-3, 6-2.

For the hard 4 or hard 10, there are 8 rolls that lose and 1 that wins the bet,
so the house wins 8 for every 7 it pays,
a gain of 1 every 9 instances, or 1/9 = 0.111111.

For the hard 6 or hard 8, there are 10 rolls that lose and 1 that wins,
so the house wins 10 for every 9 it pays,

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a gain of 1 every 11 instances, or 1/11 = 0.090909.