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Answer:
(a) The probability that at least 5 of the first 10 sites contain the given keyword is 0.0328.
(b) The probability that the search engine had to visit at least 5 sites in order to find the first occurrence of a keyword is 0.4096.
Step-by-step explanation:
(a)
Let <em>X</em> = number of sites that contains the keyword.
The probability that a site contains the keyword is, <em>p</em> = 0.20.
The number of sites visited first is <em>n</em> = 10.
The random variable <em>X</em> follows a Binomial distribution with parameter <em>n</em> and <em>p</em>.
The probability mass function of <em>X</em> is:
Compute the probability that at least 5 of the first 10 sites contain the given keyword as follows:
P (X ≥ 5) = 1 - P (X < 5)
= 1 - P (X = 0) - P (X = 1) - P (X = 2) - P (X = 3) - P (X = 4)
Thus, the probability that at least 5 of the first 10 sites contain the given keyword is 0.0328.
(b)
Let <em>Y</em> = number of sites that contains the keyword.
The probability that a site contains the keyword is, <em>p</em> = 0.20.
The random variable <em>Y</em> follows a Geometric distribution with parameter <em>p</em>.
The probability mass function of <em>Y</em> is:
Compute the probability that the search engine had to visit at least 5 sites in order to find the first occurrence of a keyword as follows:
P (X ≥ 5) = 1 - P (X ≤ 4)
Thus, the probability that the search engine had to visit at least 5 sites in order to find the first occurrence of a keyword is 0.4096.