ProbabilityHardBloom L3

Question

Three missiles $A,B$ and C whose probabilities of hitting the target are $\frac{2}{3},\frac{1}{3},\frac{2}{5}$ respectively are shot at an enemy ship simultaneously. A majority of hits is required to destroy the ship. If the ship is destroyed, then

Options

A.the probability that missile $B$ failed to hit the target is $\frac{1}{5}$
B.the probability that missile $B$ failed to hit the target is $\frac{2}{5}$
C.the probability that at least one of B or C hit the target is 1
D.the probability that at least one of the missile B or C failed to hit the target is 7/10

Solution

Sol. [Key Concept: Independent missile hits with the given probabilities; combine for the ship event.]

(b) $P = \frac{\frac{2}{3} \times \frac{2}{3} \times \frac{2}{5}}{\frac{1}{3} \times \frac{1}{3} \times \frac{2}{5} + \frac{2}{3} \times \frac{2}{3} \times \frac{2}{5} + \frac{2}{3} \times \frac{1}{3} \times \frac{3}{5} + \frac{2}{3} \times \frac{1}{3} \times \frac{2}{5}} = \frac{8}{2 + 8 + 6 + 4} = \frac{2}{5}$

(d) $P = \frac{8 + 6}{2 + 8 + 6 + 4} = \frac{7}{10}$

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