Question

A group of research proposals was evaluated by a panel of experts to decide whether or not they were worthy of funding. When these same proposals were submitted to a second independent panel of experts, the decision to fund was reversed in 40% of the cases. If the probability that a proposal is judged worthy of funding by the first panel is 0.3, what are the probabilities of these events

Answer 1

Answer:

Let A represent the event that the first group approves the proposal.

Let B represent the event that the second group approves the proposal.

We know the following:

P(A) = 0.3

P(B given A) = 1 - 0.3 = 0.7

P(not B given not A) = 1 - 0.3 = 0.7 .

A) P(approved by both groups)

= P(A and B)

= P(A)P(B given A)

= 0.3(0.7)

= 0.21 .

B) P(disapproved by both groups)

= P(not A and not B)

= P(not A)P(not B given not A)

= (1 - 0.3)(0.7)

= 0.49 .

C) This problem might be ambiguous...do you mean exactly one group or at least one group? So, I will use both interpretations.

P(approved by at least one group)

= 1 - P(disapproved by both groups)

= 1 - 0.49

= 0.51 .

P(approved by exactly one group)

= P(second group reverses the first group's decision)

= 0.4 .