A can hit a target 3 times in 6 shots, B can hit a target 2 times in 6 shots and C can hit 4 times in 4 shots. They fix a volley. What is the probability that at least 2 shots hit?
ANSWER
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Hint: Note that A, B, and C hitting the shots are independent events. Find the possibilities for hitting at least two shots and calculate the probability for independent events using \[P(A \cap B \cap C) = P(A)P(B)P(C)\].
Complete step-by-step answer: It is given that A can hit a target 3 times in 6 shots, hence, the probability of A is given as follows:
\[P(A) = \dfrac{3}{6}\]
\[P(A) = \dfrac{1}{2}............(1)\]
Let the event \[\bar A\] be the event that A does not hit the target. Then, we have:
\[P(\bar A) = 1 - \dfrac{1}{2}\]
\[P(\bar A) = \dfrac{1}{2}.............(2)\]
It is given that B can hit a target 2 times in 6 shots, hence, the probability of B is given as follows:
\[P(B) = \dfrac{2}{6}\]
\[P(B) = \dfrac{1}{3}............(3)\]
Let the event \[\bar B\] be the event that B does not hit the target. Then, we have: \[P(\bar B) = 1 - \dfrac{1}{3}\]
\[P(\bar B) = \dfrac{2}{3}............(4)\]
It is given that C can hit a target 4 times in 4 shots, hence, the probability of C is given as follows:
\[P(C) = \dfrac{4}{4}\]
\[P(C) = 1............(5)\]
Let the event \[\bar C\] be the event that C does not hit the target. Then, we have: \[P(\bar C) = 1 - 1\]
\[P(\bar C) = 0............(6)\]
In a volley, we need to find the probability that at least two shots hit the target. We have the following possibilities: Both A and B hit, C does not hit, both A and C hit, B does not hit, both B and C hit, A does not hit, all three hit the target. Hence, the required probability is given as follows:
\[P = P(A \cap B \cap \bar C) + P(A \cap \bar B \cap C) + P(\bar A \cap B \cap C) + P(A \cap B \cap C)\]
We know that A, B, and C are independent events, hence, the probability of the intersection of events can be written as the product of probabilities of individual events.
Hence, the required probability is \[\dfrac{2}{3}\].
Note: You can also find the probability that less than 2 shots hit the target and subtract the value from 1 to get the probability that at least 2 shots hit the target.