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).

(a) Probability distribution for T (geometric distribution)

The probability distribution for T follows a geometric distribution, represented by the formula:

\[ P(T = k) = (1 - p)^{k-1} \times p \]

prob_success <- 1/6

# (b) Probability that T > 3
prob_T_gt_3 <- 1 - pgeom(3, prob = prob_success)

# (c) Probability that T > 6 given T > 3
prob_T_gt_6_given_T_gt_3 <- (1 - pgeom(6, prob = prob_success)) / (1 - prob_T_gt_3)

print(prob_T_gt_3)
## [1] 0.4822531
print(prob_T_gt_6_given_T_gt_3)
## [1] 0.539031