This problem is about interrelated binomial trials.
1. In two sets of binomial trials T and t , the probabilities that a trial has a successful outcome are P and p , respectively, with corresponding probabilities of failure of Q = 1 - P and q = 1 - p . One "game" consists of a trial T , followed, if T is successful, by a trial t and then a further trial T . The two trials continue to alternate until one of the T -trials fails, at which point the game ends. The score S for the game is the total number of successes in the t -trials. Find the PGF for S and use it to show that
E [ S ] =Pp
V [ S ]=QPp (1 - Pq )
2. Two normal unbiased six-faced dice A and B are rolled alternately starting with A ; if A shows a 6 the experiment ends. If B shows an odd number no points are scored, if it shows a 2 or a 4 then one point is scored, whilst if it records a 6 then two points are awarded. Find the average and standard deviation of the score for the experiment and show that the latter is the greater.