Title: On Quaker’s Dozen

Author: Bob Schmidt

Date: 07 May 2018 17:42:11 +01:00 or Mon, 07 May 2018 17:42:11 +01:00

Summary: A student examines the Quaker’s Dozen wager.

Body: 

The Baron’s latest wager set Sir R----- the task of rolling a higher score with two dice than the Baron should with one twelve sided die, giving him a prize of the difference between them should he have done so. Sir R-----’s first roll of the dice would cost him two coins and twelve cents and he could elect to roll them again as many times as he desired for a further cost of one coin and twelve cents each time, after which the Baron would roll his.

The simplest way to reckon the fairness of this wager is to re-frame its terms; to wit, that Sir R----- should pay the Baron one coin to play and thereafter one coin and twelve cents for each roll of his dice, including the first. The consequence of this is that before each roll of the dice Sir R----– could have expected to receive the same bounty, provided that he wrote off any losses that he had made beforehand.

Put in these terms it is self evident that Sir R----- should set himself a constant goal at which to stick, for if it were to maximise his expected prize after the first roll, then it should also do so after every subsequent roll. I explained as much to the Baron, but I suspect that I did not have his undivided attention at the time.

Now, if we label Sir R-----’s roll with x_r and the Baron’s with x_b then we can therefore formulate his expected winnings as

where E[X|C] is the expected value of X given that the condition C holds true.

The probabilities that Sir R-----’s score were equal to any particular value of k are easily figured to be

from which we can trivially deduce that the probabilities that it were greater than or equal to any given value of k are

Once Sir R----- had chosen to stick with a roll of x_r, he should have expected a prize of

where Σ is the summation sign, which we can reorganise into

and so, by the very definition of conditional expectations, we have

We can make light work of figuring these expectations by noting that

[Web Editor's Note: The preceding formula has been corrected - the trailing "m" present in the PDF version has been removed.]

and proceeding backwards from twelve to two

Sir R-----’s expected winnings were therefore as shown below, and I should have consequently advised him to accept the Baron’s wager, provided that he stuck with a score of six or greater upon each roll! □

[Web Editor's Note: The preceding formula has been corrected - the correct value on the last line has been changed to "870,912" from the "870.912" in the PDF version.]

Notes: 

More fields may be available via dynamicdata ..