Question

in august 2022, the white house released a plan to forgive the payment of federal student loan for some borrowers. it was determined that this would impact nearly 30% of the total outstanding federal student loan balance.
two unlv students were randomly interviewed and said that they had outstanding federal student loan. assuming that the decision to forgive one students federal student loan is made independently from student to student:
a) what is the probability that both students will have their federal student loans forgiven? (write your final answers without rounding them.)
b) what is the probability that at least one of the two students will have her/his federal student loan forgiven?
c) what is the probability that neither student will have her/his federal student loan forgiven?

Explanation:

Step1: Define probability of loan forgiveness

Let \(p = 0.3\) (probability a single student's loan is forgiven).

Step2: Calculate probability for part (a)

Since events are independent, use \(P(A\cap B)=P(A)\times P(B)\).
For two students, \(P(\text{both})=0.3\times0.3 = 0.09\).

Step3: Calculate probability for part (b)

Use \(P(\text{at least one})=1 - P(\text{neither})\).
First find \(P(\text{neither})=(1 - 0.3)\times(1 - 0.3)=0.7\times0.7 = 0.49\).
Then \(P(\text{at least one})=1-0.49 = 0.51\).

Step4: Calculate probability for part (c)

As above, \(P(\text{neither})=(1 - 0.3)\times(1 - 0.3)=0.49\).

Answer:

a) \(0.09\)
b) \(0.51\)
c) \(0.49\)