Question

Two professors are applying for grants. ProfessorJane has a probability of 0.62 of being funded. Professor Joe hasprobability 0.25 of being funded. Since the grants are submitted totwo different federal agencies, assume the outcomes for each grantare independent.

1.What is the probability that both professors get their grantsfunded? Give your answer to four decimal places.
2.What is the probability that at least one of the professors will befunded? Give your answer to four decimal places.
3.What is the probability that Professor Jane is funded but ProfessorJoe is not? Give your answer to four decimal places.
4.Given at least one of the professors is funded, what is theprobability that Professor Jane is funded but Professor Joe is not?Give your answer to four decimal places.

Answer 1

1. P(both Professors funded)=0.155

2. P(at least one Professor funded)=0.715

3. P(Jane funded but Joe not funded)=0.465

4. P(Professor Jane is funded but Professor Joe is not/at least one Professor funded)=0.6503

Explanation:

P(Professor Jane funded)=P(F1)=0.62

P(Professor Joe funded)=P(F2)=0.25

P(F1')=1-P(F1)=1-0.62=0.38

P(F2')=1-P(F2)=1-0.25=0.75

1.

P(both Professors funded)=P(F1∩F2)

P(both Professors funded)=P(F1)*P(F2)

P(both Professors funded)=0.62*0.25

P(both Professors funded)=0.155

2.

P(at least one Professor funded)=P(F1∪F2)

P(at least one Professor funded)=P(F1)+P(F2)-P(F1∩F2)

P(at least one Professor funded)=0.62+0.25-0.155

P(at least one Professor funded)=0.87-0.155

P(at least one Professor funded)=0.715

3.

P(Jane funded but Joe not funded)=P(F1∩F2')

P(Jane funded but Joe not funded)=P(F1)*P(F2')

P(Jane funded but Joe not funded)=0.62*0.75

P(Jane funded but Joe not funded)=0.465

4.

P(Professor Jane is funded but Professor Joe is not/at least one Professor funded)=P(F1∩F2')/P(F1∪F2)

=P(F1∩F2')∩(F1∪F2)/P(F1∪F2)

=P(F1∩F2')/P(F1∪F2)

P(Professor Jane is funded but Professor Joe is not/at least one Professor funded)=0.465/0.715

P(Professor Jane is funded but Professor Joe is not/at least one Professor funded)=0.6503