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

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.