Upon meeting at a gambling establishment, CR and I got into a dispute over the probability of a certain event. CR believed that a particular slot machine paid off 1 time out of 3, when it flickered a blue light, and 3 times out of 7 when it was illuminated pink. I argued that it paid on average 3 times out of 8, no matter what light was on. We both agreed, however, that the blue light is on 9/16 of the time, while pink glows the remaining 7/16 of the time.

The dispute raged on, and neither of us could be swayed by the other's argument. Exasperated, CR offered to beat me with the whiteboard bearing his calculations. But I suggested a wager instead:

We'd set the odds as CR calculated; and I would choose which side to bet on. Thus we'd set our betting odds on "machine payoff" 1 ruble against 2 for a blue light and 3 rubles against 4 for a pink light. And I would, naturally, bet on a "payoff" in the case of "blue", and against a "payoff" in the case of "pink."

This being a popular slot machine and many casino guests waiting in line to get to it, we’d have no shortage of statistical samples.

"This way, each of us can stick to their opinion, but at the end of the day one of us will walk away richer," says I.

Is that bet a fair resolution for the dispute (statistically)? If not, how is it uneven?

Assume for the purposes of this problem that one of us is right and another is wrong (not both wrong). Make no assumptions regarding our respective faculties to estimate probabilities.

## Question

## Prime 15

This probability problem was born of this forum.

Upon meeting at a gambling establishment, CR and I got into a dispute over the probability of a certain event. CR believed that a particular slot machine paid off 1 time out of 3, when it flickered a blue light, and 3 times out of 7 when it was illuminated pink. I argued that it paid on average 3 times out of 8, no matter what light was on. We both agreed, however, that the blue light is on 9/16 of the time, while pink glows the remaining 7/16 of the time.

The dispute raged on, and neither of us could be swayed by the other's argument. Exasperated, CR offered to beat me with the whiteboard bearing his calculations. But I suggested a wager instead:

We'd set the odds as CR calculated; and I would choose which side to bet on. Thus we'd set our betting odds on "machine payoff" 1 ruble against 2 for a blue light and 3 rubles against 4 for a pink light. And I would, naturally, bet on a "payoff" in the case of "blue", and against a "payoff" in the case of "pink."

This being a popular slot machine and many casino guests waiting in line to get to it, we’d have no shortage of statistical samples.

"This way, each of us can stick to their opinion, but at the end of the day one of us will walk away richer," says I.

Is that bet a fair resolution for the dispute (statistically)? If not, how is it uneven?

Assume for the purposes of this problem that one of us is right and another is wrong (not both wrong). Make no assumptions regarding our respective faculties to estimate probabilities.

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