Respuesta :
Answer and explanation:
Given : Assume that each of your calls to a popular radio station has a probability of 0.02 of connecting, that is, of not obtaining a busy signal. Assume that your calls are independent.
This form an geometric probability,
Probability that call connects p=0.02
Probability that call does not connect q=1-0.02=0.98
To find :
(a) What is the probability that your first call that connects is your 10th call?
The formula is given by,
[tex]P(X=k)=q^{k-1}p[/tex]
[tex]P(X=10)=(0.98)^{10-1}\times 0.02[/tex]
[tex]P(X=10)=(0.98)^{9}\times 0.02[/tex]
[tex]P(X=10)=0.01667[/tex]
(b) What is the probability that it requires more than five calls for you to connect?
[tex]\small P = P(X>5) = 1 - P(X \leq 5)[/tex]
[tex]\small So,P(X>5) = 1 - P(X=1)- P(X=2)- P(X=3)- P(X=4)-P(X=5)[/tex]
[tex]\tiny \Rightarrow P(X>5)=1 - (0.98)^{0}(0.02) - (0.98)^{1}(0.02) -(0.98)^{2}(0.02)-(0.98)^{3}(0.02) - (0.98)^{4}(0.02)[/tex]
[tex]\tiny \Rightarrow P(X>5)=1 - 0.02 - 0.0196 -0.019208-0.01882384 - 0.0184473632[/tex]
[tex]\tiny \Rightarrow P(X>5)=0.90392[/tex]
(c) What is the mean number of calls needed to connect?
The mean value of a geometric distribution with probability of success = p is given by -
[tex]\mu = E[X] = \frac{1}{p}[/tex]
[tex]\mu = E[X] = \frac{1}{0.02}[/tex]
[tex]\mu = E[X] =50[/tex]
