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17 June, 12:07

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

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  1. 17 June, 13:15
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    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.
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