Question

Susan is taking Western Civilization this semester on a pass/fail basis. The department teaching the course has a history of passing 84% of the students in Western Civilization each term. Let n = 1, 2, 3, ... represent the number of times a student takes Western Civilization until the first passing grade is received. (Assume the trials are independent.)
(a) Write out a formula for the probability distribution of the random variable n. (Use p and n in your answer.)
(b) What is the probability that Susan passes on the first try (n = 1)? (Use 2 decimal places.)
(c) What is the probability that Susan first passes on the second try (n = 2)? (Use 3 decimal places.)
(d) What is the probability that Susan needs three or more tries to pass Western Civilization? (Use 3 decimal places.)
(e) What is the expected number of attempts at Western Civilization Susan must make to have her (first) pass? Hint: Use μ for the geometric distribution and round.

Answer 1

Answer:

a) $P(x=n)=(1-p)^{n-1}p$

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:

$P(x=n)=(1-p)^{n-1}p$

(b) What is the probability that Susan passes on the first try (n = 1)?

We can calculate this with the previous expression

$P(x=1)=(1-p)^{1-1}p=p=0.84$

(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.

$P(x=2)=(1-p)^{2-1}p=(1-p)p=0.16*0.84=0.134$

(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.

$P(x\geq 3)=1-(P(1)+P(2))=1-(0.84+0.134)=1-0.974=0.026$

(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:

$E(n)=\frac{1}{p} =\frac{1}{0.84}= 1.19$

The expected number of attemps is 1.19.