Probability Question Puzzle

Alice, who has never formally studied probability theory, wonders what the probability of a score of seven from throwing two dice is. (Maybe she has been invited to gamble on it.) She asks a friend, Bob who is a consultant. He says “dunno, are the dice fair?”. Alice says “yes”. Bob consults a reputable maths book which says:

The chance of two fair dice scoring 7 is 1/6.

(To see this, note that whatever is on the first of the dice, there is a 1/6 chance that the second will give as sum of 7.)

Bob replies “1/6”. This seems pragmatic, but is it actually correct? Any quibbles?
(As this is a blog on uncertainty, you may be sure there are.)
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If Alice is asking about idealised ‘mathematical dice’, then Bob’s answer is correct. Bob might more pedantically say, “If the dice are really fair, the probability is 1/6”, which draws attention to Alice’s belief. But will she appreciate the subtleties here? (Does Bob?.)

If Bob is a mathematician, then Bob will realise that Alice can’t possibly know for sure that any actual dice are fair, and so can’t know that the required probability is actually 1/6. Bob might not think that ‘the dice are fair’ is meaningful, and so might not think that any straightforward answer was meaningful or ‘operationalizable’.

To give a mathematical answer Bob would need to ‘mathematize’ the situation. He might make a first attempt in terms of ‘belief’ or ‘expectation’. Bob might say “If you think the dice are fair, then ‘rationally’ you should think that the required probability is 1/6.” But would Bob be justified in promoting such ‘rationality’, and would Alice be justified in acting on it? Alternatively, Bob might say “Given that you think the dice fair, you should expect a probability of 1/6.” But might not confuse this real-world expectation with the psychologists’ ‘mathematical expectation’?

The difficulty here is that these approaches assume that Alice is thinking in ways that don’t make much sense, logically. An alternative approach is as follows:

If Alice thinks that the dice are ‘probably’ fair, she presumably has some grounds for such a view. Presumably, then, the dice have passed some tests for fairness. If ‘the dice’ are unaltering, then a reasonable ‘inductive’ heuristic is to ‘expect’ them to pass the same tests again (subject to the usual statistical considerations).

In this case, it might be reasonable to think that a test of fairness would either directly or indirectly check on the propensity of scoring 7. It would then be a reasonable application of a reasonable heuristic to suppose that the dice will continue to pass the test. Thus Bob might reasonably say that the heuristic answer is 16/, and that he is not aware of any specific reason to doubt this. But maybe this propensity hasn’t been adequately tested? Or maybe Alice is just relying on some kind of ‘principle of indifference’, and the propensity hasn’t been tested at all? In this case, there is greater uncertainty. But is it Bob’s responsibility to assess and communicate this?

It seems to me that effective communication between Alice and Bob would rely on some pre-established understanding of the above issues.

See Also

Other puzzles (not all as ‘deep’ as this).

Dave Marsay

 

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