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# A is targeting to B, B and C are targeting to A. Probability of hitting the target by A, B and C are 2/3, 1/2 and 1/3 respectively. If A is hit then find the probability that B hits the target and C does not.

6 years ago
Hello Student,
Let us consider the events
E1≡ A hits B Then P (E1) = 2/3
E2≡ B hits A P (E2) = 1/2
E3≡ C hits A P (E3= 1/3
E ≡ A is hit
P (E) = P (E2∪ E3) = 1 – P ($\underset{E}{\rightarrow}$2$\underset{E}{\rightarrow}$3)
= 1 P ($\underset{E}{\rightarrow}$2) P ($\underset{E}{\rightarrow}$3) = 1 – 1/2. 2/3 = 2/3
To find P (E2$\underset{E}{\rightarrow}$2/ E)
= P (E2$\underset{E}{\rightarrow}$3)/P (E) [∵ P (E2$\underset{E}{\rightarrow}$3∩ E) = P (E2$\underset{E}{\rightarrow}$3) i.e., B hits A is hit = B hits A].
= P (E2). P ($\underset{E}{\rightarrow}$3)/P (E) = 1/2 X 2/3 / 2/3 = 1/2

Thanks