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.
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
(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
(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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