Respuesta :
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 X = number of sites that contains the keyword.
The probability that a site contains the keyword is, p = 0.20.
The number of sites visited first is n = 10.
The random variable X follows a Binomial distribution with parameter n and p.
The probability mass function of X is:
[tex]P(X=x)={10\choose x}0.20^{x}(1-0.20)^{10-x};\ x=0,1,2,3...[/tex]
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)
[tex]=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[/tex]
Thus, the probability that at least 5 of the first 10 sites contain the given keyword is 0.0328.
(b)
Let Y = number of sites that contains the keyword.
The probability that a site contains the keyword is, p = 0.20.
The random variable Y follows a Geometric distribution with parameter p.
The probability mass function of Y is:
[tex]P(Y=y)=(1-p)^{x-1}p;\ x=1,2,3...[/tex]
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)
[tex]=1-\sum\limits^{4}_{x=1} (1-0.20)^{x-1}0.20\\=1-0.20-0.16-0.128-0.1024\\=0.4096[/tex]
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.
