Question

An internet search engine looks for a certain keyword in a sequence of independent websites. It is believed that 35% of the sites contain this keyword.

(a) Let X be the number of websites visited until the first keyword is found. Compute the probability that the search engine had to visit at least 10 sites in order to find the first occurrence of the keyword.

(b) Out of the first 25 websites, let Y be the number of sites that contain the keyword. Compute the probability that at least 10 of the first 25 websites contain the keyword.

Answer 1

p = 0.35

a) X ~ G(0.35)

P(X = x) = (1 - 0.35)^x-1 * 0.35

P(X > 10) = 1 - (P(X = 1) + P(X = 2) + P(X = 3) + ... + P(X = 9))

                 = 1 - (0.65⁰ * 0.35 + 0.65¹ * 0.35 + 0.65² * 0.35 +... + 0.65⁸ * 0.35 )

                 = 1 - 0.9793

                 = 0.0207

b) n = 25

p = 0.35

This is a binomial distribution.

P(X = x) = 25Cx * 0.35^x * (1 - 0.35)^25-x

P(X > 10) = 1 - (P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + ... + P(X = 9))

                = 1 - (25C0 * 0.35⁰ * 0.65²⁵ + 25C1 * 0.35¹ * 0.65²⁴ + 25C2 * 0.35² * 0.65²³ + ... + 25C0 * 0.35⁹ * 0.65¹⁶ )

                = 1 - 0.6303

                = 0.3697