Let X = number of successes, i.e. number of odd numbers.
p = probability of getting an odd number in a single throw of a die
The p.m.f. of X is given by
Hence, the probability of 5 successes is 3/32.
(ii) P(at least 5 successes) = P[X ≥ 5]
= p(5) + p(6)
Hence, the probability of at least 5 successes is 7/64.
(iii) P(at most 5 successes) = P[X ≤ 5] = 1 – P[X > 5]
Hence, the probability of at most 5 successes is 63/64.