P,Q,R shot to hit a target. If P hits it 3 times in 4 trials, Q hits it 2 times in 3 trials and R hits it 4 times in 5 trials, what is the probability that the target is hit by at least two persons?

Let E1, E2 and E3 be independent events that P, Q and R hit the target respectively. Then
P(E1) = 3/4, P(E2) = 2/3and P(E3) = 4/5
P(E1c) =1 - 3/4 = 1/4, P(E2c) =1 - 2/3 = 1/3and P(E3c) = 1 - 4/5 = 1/5 .

The target is hit by atleast two persons in the following mutually exclusiveways:

(a) P hits, Q hits and R does not hit i.e., E1 E2E3c
(b) P hits, Q does not hit and R hits i.e., E1 E2cE3
(c) P does not hit, Q hits and R hits i.e., E1c E2E3
(d) P hits, Q hits and R hits i.e., E1 E2E3
Required Probability = P[(a) (b) (c) (d)] = P(a) + P(b) + P(c) + P(d)

= (3/4 x 2/3 x 1/5) + (1/4 x 2/3 x 4/5) + (3/4 x 2/3 x 4/5)

= 25/30 = 5/6.