Question

Can you please do them You are planning to take three exams. According to the records, the failure rates for the exams A, B, and C are 15%, 25%, and 35%, respectively. Assume that the passing rates for all exams are independent events:

1. What will be the probability that you will pass all three exams?

2. What is the probability that you will pass at least two exams?

3. What is the probability that you will pass at least one exam?

4. Given that you have passed at least one of the exams, what is the probability that you have passed only one exam?

Answer 1

failure rates for A is 15% . so success rate is 85%.

failure rates for B is 25% . so success rate is 75%.

failure rates for C is 35% . so success rate is 65% .

let , success events in A , B , C have probabilities P(A),P(B),P(C)

Q1.

probability of passing all exams is p(A and B and C)

=P(A)P(B)P(C)

=0.85*0.75*0.65

=0.414375

Q2.

probability of passing at least 2 exams

=P(A)P(B)P(C)+P(A)P(B)P(C^c)+P(A^c)P(B)P(C)+P(B^c)P(A)P(C)

=0.85*0.75*0.65+0.85*0.75*0.35+0.15*0.75*0.65+0.25*0.85*0.65

=0.84875

Q3.

probability of passing in atleast one exam

=1-probability of failing in all three

=1-P(A^C)P(B^C)P(C^c)

=1-0.15*0.25*0.35

=1-0.013125

=0.986875

Q4.

probability of passing only one exam , given that passed at least one)

=prob(passing only one)/prob(passing atleast one )

now , probability of passing only one exam

=P(A)P(B^C)P(C^C)+P(A^C)P(B^C)P(C)+P(B)P(A^C)P(C^C)

=0.85*0.25*0.35+0.15*0.25*0.65+0.15*0.35*0.75

=0.138125

so , required probability

=0.138125/0.986875

=0.1399