7 . A die is rolled until the first time T that a six turns up. (a) What is the probability distribution for T? (b) Find P(T > 3). (c) Find P(T > 6|T > 3)
The probability distribution for \(T\), where \(T\) is the trial on which the first six appears, follows a geometric distribution with success probability \(p = 1/6\).
#you have a 5 in 6 chance of NOT getting the 6 each time.
p_t_gt_3 <- (5/6)^3
p_t_gt_3## [1] 0.5787037#giving you the probability of not getting the six in the next 3 tries, given that you haven't gotten it in the first 3 tries.
p_t_gt_6_given_t_gt_3 <- (5/6)^6 / (5/6)^3
p_t_gt_6_given_t_gt_3## [1] 0.5787037