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grandymaker [24]
3 years ago
7

An internet search engine looks for a certain keyword in a sequence of independent web sites. It is believed that 20% of the sit

es contain this keyword. (a) Compute the probability that at least 5 of the first 10 sites contain the given keyword. (b) Compute the probability that the search engine had to visit at least 5 sites in order to find the first occurrence of a keyword.
Mathematics
1 answer:
marusya05 [52]3 years ago
4 0

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:

P(X=x)={10\choose x}0.20^{x}(1-0.20)^{10-x};\ x=0,1,2,3...

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)

              =1-\sum\limits^{4}_{x=0} {10\choose x}0.20^{x}(1-0.20)^{10-x}\\=1-0.1074-0.2684-0.3020-0.2013-0.0881\\=0.0328

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:

P(Y=y)=(1-p)^{x-1}p;\ x=1,2,3...

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)

             =1-\sum\limits^{4}_{x=1} (1-0.20)^{x-1}0.20\\=1-0.20-0.16-0.128-0.1024\\=0.4096

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.

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