# Probability Coupled with Payout

My brother and I made a bet which I took. He had two coffee cups which were roll up the rim to win. The odds of a cup winning were 1 in 6. He was feeling very confident and said "if I don't win with both of these cups I pay your a dollar, but if I do win with both of these cups you pay me 10 dollars".

I understand that the probability of both cups winning is 1/6 * 1/6 = 1/36, which is the basis of why I took the bet. But how does the payout ratio affect this bet?

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Your expected return from the bet is $\frac {35}{36}\cdot 1 + \frac 1{36} \cdot (-10)=+\frac {25}{36}$ so the bet is in your favor. If you won $1$ but had to pay $40$ the return would be $\frac {35}{36}\cdot 1 + \frac 1{36} \cdot (-40)=-\frac {5}{36}$ and the bet would be unfavorable. You would still win most of the time, but if you made the bet a large number of times you would lose in total.