MATH (Probability)

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Jackson and Alice are the boss and secretary respectively working in an office for a firm. At
any particular time during office hours, the probability that Jackson is in the office is 0.82 and
probability that Alice is in the office is 0.98. The probability that Jackson is in the office given
that Alice is in the office is 0.76. Find the probability that at any particular time during office
hours,

i)Both Jackson and Alice are in the office.

ii)Alice is the office given that Jackson is in the office.

iii)At least one of them is in the office.

  • MATH (Probability) -

    The general definition of "conditonal probablilty" is
    P(A | B) = P(A ∩ B)/P(b) , the left side read as "the probability of A given B

    P(J) = .82
    P(A) = .98

    P(J|A) = P(J∩A)/P(A)
    .76 = P(J∩A)/.82

    i) P(J∩A) = .6232
    ii) P(A|J) = P(A∩B)/P(J) = .6232/.82 = .76

    iii) prob at least one of them in the office ---> P(A∪B)
    = P(A) + P(B) - P(A∩B)
    = .98 + .82 - .6232 > 1

    Unless a made an arithmetic error, the data appears to be flawed.

  • MATH (Probability) -

    A squadron of 16 bombers needs to penetrate air defenses to reach its target. They can
    either fly low and expose themselves to the air defense guns, or fly high and expose themselves to
    surface-to-air missiles. In either case, the air defense firing sequence proceeds in three stages. First,
    they must detect the target, then they must acquire the target (lock on target), and finally they
    must hit the target. Each of these stages may or may not succeed. The probabilities are as follows:
    The gums can fire 20 shells per minute, and the missile installation can fire three per minute. The
    AD Type Pdetect Pacquire Phit
    Low 0.90 0.80 0.05
    High 0.75 0.95 0.70
    proposed flight path will expose the planes for one minute if they fly low, and five minutes if they
    fly high.
    (a) Determine the optimal flight path (low or high). The objective is to maximize the number of
    bombers that survive to strike the target.
    (b) Each individual bomber has a 70% chance to destroy the target. Use the results of part (a)
    to determine the chances of success (target destroyed) for this mission.
    (c) Determine the minimum number of bombers necessary to guarantee a 95% chance of mission
    success.
    (d) Perform a sensitivity analysis with respect to the probability p = 0.7 that an individual
    bomber can destroy the target. Consider the number of bombers that must be sent to
    guarantee a 95% chance of mission success.
    (e) Bad weather reduces both Pdetect and p, the probability that a bomber can destroy the target.
    If all of these probabilities are reduced in the same proportion, which side gains an advantage
    in bad weather?

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