Let X = number of winning prizes.
p = probability of winning a prize
The p.m.f. of X is given by
(i) P(a person wins a prize at least once)
Hence, probability of winning a prize at least once = 1 – \(\left(\frac{99}{100}\right)^{50}\)
(ii) P(a person wins exactly one prize) = P[X = 1] = p(1)
Hence, probability of winning a prize exactly once = \(\frac{1}{2} \left(\frac{99}{100}\right)^{49}\)
(iii) P(a persons wins the prize at least twice) = P[X ≥ 2]
= 1 – P[X < 2]
= 1 – [p(0) + p(1)]
Hence, the probability of winning the prize at least twice = 1 – 149 \(\left(\frac{99^{49}}{100^{50}}\right).\)