Consider an eight-sided die numbered 1 to 8 which is loaded in such a way that the probability of each face turning up is proportional to the number of dots on that face. For example, a six is three times as probable as a two.

The die is rolled until a run of 8 different faces appears. For example, one might roll the sequence 6565742726486467483472516 with only the last eight rolls all distinct.

(i) What is the expected number of rolls for the above event?

(ii) How will the answer change if the last 8 rolls was required to produce 12345432 (strictly in this order?

(In reply to Attempt at simulating part ii by Charlie)

36^8/(1*2*3*4*5*4*3*2)  is approximately 979552051 and is probably an underestimate of the number of rolls needed for part ii.

Posted by Charlie on 2016-09-08 13:56:17