Question Number 102142 by Ar Brandon last updated on 06/Jul/20

A four-sidex dice with numbered 1, 2, 3, and 4 is thrown  and the number at the base is read.  The dice is biased such that the probabilies P_1 , P_2 , P_3 ,  and P_4  to obtain 1, 2, 3, and 4 respectively are in an  arithmetic progression.  1\ Given P_4 =0.4, calculate P_1 , P_2 , and P_3 .  2\ The dice is thrown n-times (n≥1). The throws are assumed  to be independent, 2 by 2, and identical. Given U_n -the probability  of obtaining for the first time the fourth-n^(th) throw;  a\Express U_n  in terms of n=  b\Given S_n =Σ_(i=1) ^n U_i     i. Express Sn in terms of n, and find its limit.   ii. Determine the smallest natural number such that        S_n >0.999