Probability Question

Inconsequential

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Apr 17, 2021
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The Question is as follows:
Mr. A believes that the Democrat will be elected in a presidential election with probability 5/8. Mr.B believes the Republican will be elected with probability 3/4. Neither gives third party candidates any chance at all. They agree to bet $10 on the outcome at even odds. (Thus Mr.A will pay Mr.B $10 if the Republican wins, and Mr.B will pay him $10 if the Democrat wins.)
  • What is Mr.A’s expected dollar gain?
  • What is Mr.B’s?
  • How would you be able to make money for sure by betting with Mr.A and Mr.B if they are both always ready to accept any bet that they believe has a nonnegative dollar expectation?
My attempt so far:
According to Mr.A, Democrat will win with probability 5/8, it means probability of a republican win is 3/8 in his case.
According to Mr.B, Republican will win with probability 3/4, it means probability of a democrat win is 1/4 in his case.
will it make sense if probabilities of democrat (and republican) win in both cases are added and find a solution.
 

Dr.Peterson

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Nov 12, 2017
Messages
11,268
The Question is as follows:
Mr. A believes that the Democrat will be elected in a presidential election with probability 5/8. Mr.B believes the Republican will be elected with probability 3/4. Neither gives third party candidates any chance at all. They agree to bet $10 on the outcome at even odds. (Thus Mr.A will pay Mr.B $10 if the Republican wins, and Mr.B will pay him $10 if the Democrat wins.)
  • What is Mr.A’s expected dollar gain?
  • What is Mr.B’s?
  • How would you be able to make money for sure by betting with Mr.A and Mr.B if they are both always ready to accept any bet that they believe has a nonnegative dollar expectation?
My attempt so far:
According to Mr.A, Democrat will win with probability 5/8, it means probability of a republican win is 3/8 in his case.
According to Mr.B, Republican will win with probability 3/4, it means probability of a democrat win is 1/4 in his case.
will it make sense if probabilities of democrat (and republican) win in both cases are added and find a solution.
You have to look at it from each person's perspective, separately.

Using Mr. A's assumption about the probability, what is his expectation on this bet? (That is, literally, what does he expect?) Then do the same with Mr. B's assumption. None of this is about what the actual probabilities are, or what you might think they are.

Then you can think about the third question.
 

Inconsequential

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Apr 17, 2021
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You have to look at it from each person's perspective, separately.

Using Mr. A's assumption about the probability, what is his expectation on this bet? (That is, literally, what does he expect?) Then do the same with Mr. B's assumption. None of this is about what the actual probabilities are, or what you might think they are.

Then you can think about the third question.
Thank you for your response.
I think, one way to determine the expected dollar gain for both persons separately is to use this formula.
(Amount won * probability of winning) – (Amount lost * probability of losing)
But if it is not about the actual probabilities then I am going in the wrong direction.
 

Inconsequential

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Another way to solve it may be through expected utility calculation.

u = probability_1(first outcome) + probability_2 (second outcome)
u = 0.625(10) + 0.375(-10)
u = 6.25 - 3.75 = 2.5 -------expected dollar gain for Mr.A

u = 0.75(10) - 0.25(-10)
u = 7.5 - 2.5 = 5 ------expected dollar gain for Mr.B
 

Dr.Peterson

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Joined
Nov 12, 2017
Messages
11,268
I think, one way to determine the expected dollar gain for both persons separately is to use this formula.
(Amount won * probability of winning) – (Amount lost * probability of losing)
Another way to solve it may be through expected utility calculation.

u = probability_1(first outcome) + probability_2 (second outcome)
These are really the same thing; both are expected value.

u = 0.625(10) + 0.375(-10)
u = 6.25 - 3.75 = 2.5 -------expected dollar gain for Mr.A

u = 0.75(10) - 0.25(-10)
u = 7.5 - 2.5 = 5 ------expected dollar gain for Mr.B
Correct; so each thinks his bet is a good one.

Now, what bets would each of them make? Is there a way to make a bet with each of them that they would accept, such that you would come out ahead no matter which party wins?
 
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