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Probability : Dice and Payoff

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Suppose you roll two dice
a) What are the odds in favor of rolling a sum of 10?
b) What are the odds against rolling a sum of 7 or 11?
c) In part a, if you bet one dollar that a sum of 10 will turn up, what should the house pay (plus returning your one dollar bet) if a sum of 10 turns up for the game to be fair?

https://brainmass.com/math/probability/probability-dice-payoff-10292

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a) Let x=number of points on the first die and
y=number of points on the second die.

Then P(x+y=10)=3/36=1/12
since we have 3 wining possibilities for x+y=10 (x=4,y=6 or x=5,y=5 or x=6,y=4) out of 36 (the total number of outcomes when you roll 2 dice).

b) From the same reason we have P(x+y=7)=6/36 and P(x+y=11)=2/36
Therefore P(x+y is neither 7 nor 11)=1-P(x+y=7)-P(x+y=11)=1-6/36-2/36=28/36

c) In part a) your chances to win are 3/36 and your chances to loose are 1-3/36=33/36.
So to loose there are 11 times more chances to loose than to win.(compare 33/36 to 3/36)
So in order the game to be fair, if you bet 1 dollar, if you win the house should pay you 11 dollars plus your dollar.

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