Answer:
a) [tex]P(x=n)=(1-p)^{n-1}p[/tex]
b) P(1)=0.84
c) P(2)=0.134
d) P(n≥3)=0.026
e) The expected number of attemps is 1.19.
Step-by-step explanation:
We use the geometric distribution to model this random variable.
(a) Write out a formula for the probability distribution of the random variable n.
The expression for the probability of "times a student has to take Western Civilization" is:
[tex]P(x=n)=(1-p)^{n-1}p[/tex]
(b) What is the probability that Susan passes on the first try (n = 1)?
We can calculate this with the previous expression
[tex]P(x=1)=(1-p)^{1-1}p=p=0.84[/tex]
(c) What is the probability that Susan first passes on the second try (n = 2)?
This probability considers she failed the first try, and passes on the second try.
[tex]P(x=2)=(1-p)^{2-1}p=(1-p)p=0.16*0.84=0.134[/tex]
(d) What is the probability that Susan needs three or more tries to pass Western Civilization? (Use 3 decimal places.)
This is equal to the probability of passing in the first or second try.
[tex]P(x\geq 3)=1-(P(1)+P(2))=1-(0.84+0.134)=1-0.974=0.026[/tex]
(e) What is the expected number of attempts at Western Civilization Susan must make to have her (first) pass?
The expected value for passing Western Civilization is:
[tex]E(n)=\frac{1}{p} =\frac{1}{0.84}= 1.19[/tex]
The expected number of attemps is 1.19.
