A pair of dice, when rolled, produces sums of 2 to 12, with varying probabilities. Can the dice be reweighted (each face assigned a probability other than 1/6) so that all 11 sums occur with the same frequency?
If so how, if not how close can the difference between the least and most likely sum be made?

(In reply to re(2): A further improvement by Jer)

With the result that the ratio is at least 2, we can show my weightings hits a lower bound and is thus optimal.

P(sum=2) = P(sum=12)= p
P(sum=7) = 2p

The difference is at least p so we seek to minimize p.

It would be higher if one of the other eight sums had a probability less than p or greater than 2p.  So we set them all at 2p.

The probability of each of the nine sums from 3 to 11 is 2p, for a grand total of 20p.

Being a probability distribution this must equal 1.
20p=1
p=1/20

So the minimum difference is 1/20. 

Posted by Jer on 2011-01-24 16:44:44