The probability of A hitting a target is #2/3# and that of B is #4/5#. They both fire at the target. What is the probability that at best one of them hits the target? What is the probability that only one of them will hit the target?

1 Answer
MattyMatty
Dec 6, 2017

#Part 1 : 7/15 ; Part 2 : 2/5#

Explanation:

#P[ "at best one of them hits the target "]#
= #1 - P[" both hit the target"]#
= #1 - (2/3)(4/5)#
= #1 - 8/15#
= #7/15#

#P[ "only one of them will hit the target" ]#
= #P["A hits target XOR B hits target"]#
(XOR = exclusive or)
= #P["A hits target"] + P["B hits target"] - 2 P["A AND B hit"]#
= #2/3 + 4/5 - 2(2/3)(4/5)#
= #(10 + 12 - 16)/15#
= #6/15#
= #2/5#

Note that the formula for OR is
P[A or B] = P[A] + P[B] - P[A AND B]
But the formula for XOR is
P[A xor B] = P[A] + P[B]- 2 P[A AND B].

The whole calculation is under the assumption that both shooters shoot #"independent"# of each other so that
#P["A AND B"] = P[A] * P[B]#.

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