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The Backgammon Solution
Posted By: Timothy Chow
Date: Sunday, 16 October 2016, at 6:41 p.m.
In Response To: The Backgammon Solution (Rick Janowski)
The actual value of Q does make a difference because it affects the relative probabilities of the two possibilities.
Suppose you knew, for example, that the people who decided how much money to give for the prize rolled a die and multiplied the value of the die by £1000 to obtain the smaller prize, and then doubled that to obtain the larger prize. Then what you should do is switch if Q ≤ 6000 and stay if Q ≥ 8000.
In the problem as stated, of course, you don't know how the amount of money was determined. But it doesn't follow that, just because you don't know, that you can rationally hypothesize that the £1000–£2000 case and the £2000–£4000 case are equally likely.
What you can say about equal likelihood is that if you're in the 1000–2000 case then you're equally likely to see £1000 or £2000, and similarly that if you're in the 2000–4000 case then you're equally likely to see £2000 or £4000. But this is not the same as saying that the £1000–£2000 case and the £2000–£4000 cases are equally likely. To assess the relative likelihood of those two cases, you have to bring in some assumption about how likely the top prize is to be £2000 in the first place. And that probability does depend on the actual value of Q, because there's no way to make all possible values of Q equally likely.

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