Probability of Target Being Hit by Either A or B

The probability of A hitting the target is 3/5 and for B it is 4/7. This article discusses the probability that the target will be hit when both attempt to hit it, given the various possibilities as to how their hit probabilities may be related.

Relationship Between Hit Probabilities

The exact relationship between the hit probabilities of A and B is crucial for calculating the probability that the target will be hit. We will consider three possible scenarios:

A and B supplement each other. A and B share a common subset. A and B are independent of each other.

Calculation of Target Hitting Probability

To calculate the probability that the target will be hit by either A or B, we can use the complementary probability method. This involves calculating the probability that both miss, and then subtracting that from 1.

Hitting Probabilities and Their Complements

First, we calculate the individual probabilities that each person misses the target:

Probability that A misses the target: Probability that B misses the target:

Given the probabilities:
P(A hits) 3/5, P(B hits) 4/7

P(A misses) 1 - P(A hits) 1 - 3/5 2/5

P(B misses) 1 - P(B hits) 1 - 4/7 3/7

Considering Independence of A and B

Since A and B are independent, the probability that both miss is the product of their individual probabilities of missing:

P(both miss) P(A misses) * P(B misses) (2/5) * (3/7) 6/35

Therefore, the probability that at least one of them hits the target is:

P(at least one hits) 1 - P(both miss) 1 - 6/35 29/35

Conclusion

The probability that the target will be hit by either A or B is 29/35.

Complementary Probability Example

For a different scenario, consider the probabilities P(A hits) 1/4 and P(B hits) 2/5. Using the complementary probability method:

Probability that neither A nor B hits: Probability that at least one of A or B hits:

P(neither A nor B hits) (1 - 1/4) * (1 - 2/5) 3/4 * 3/5 9/20

P(at least one hits) 1 - P(neither hits) 1 - 9/20 11/20

Hence, the probability that either A or B hits the target when both shoot is 11/20.