On Sat, 2 Apr 2005 13:30:11 -0500, "Brian M. Scott" <firstname.lastname@example.org> wrote:
>On Sat, 02 Apr 2005 09:38:07 -0800, Bob <email@example.com> >wrote in <news:firstname.lastname@example.org> >in alt.math.undergrad: > >> On 1 Apr 2005 23:13:09 -0800, "Joseph A." >> <email@example.com> wrote: > >>> You have up to 3 rolls of an unbiased six-sided dice, and >>> your aim is to obtain the highest result from a single >>> roll of the dice. After the first and second rolls of >>> the dice you can choose to stop (and accept the result), >>> or you can choose to roll again. What is the optimal >>> strategy and its expected payoff? > >> "optimal strategy" implies weighing pro and con for alternatives. > >No, the optimal strategy is the strategy with the highest >expected value. >
?? which means the same thing, as you show below.
>> What is the cost of doing one more roll? You have not >> stated any cost, > >Although it isn't stated explicitly, it's clear that your >final score is the *last* number that you roll,
I agree that seems likely (and thus I agree with your analysis below). But I'm not so sure it is really clear from the statement of the question; it certainly did not seem clear to the OP.
Most importantly, I think... we agree that determining the strategy requires a clear understanding of the "rules". If the OP was not clear what the game was, then it would not be possible to work out the problem. Step 1 is to be sure what the game is, either because it is stated, or because one makes an assumption.
>so the cost >of rolling again is the probability of getting a result >lower than your last roll.
yep. You are now doing what I suggested at the start, considering the con of one approach.
> >> so it would always be of advantage to >> roll again (unless you already have a 6). > >This is obviously not true. Suppose that your first roll is >a 1; clearly you cannot lose by rolling again. Suppose that >you now roll a 5. If you roll a third time, the probability >is 2/3 that you will end up with a lower score, 1/6 that you >will end up with the same score, and 1/6 that you will >improve your score; clearly you should not roll a third >time. > >Brian