To find the probability that the target is hit by P or Q but not by R, let's define the events: E1:P hits the target. E2:Q hits the target. E3:R hits the target. The given probabilities are: P(E1)=
2
3
P(E2)=
3
5
P(E3)=
5
7
These events are independent, so: The probability that P hits and both Q and R miss is: P(E1∩E2∩E3)=P(E1)⋅(1−P(E2))⋅(1−P(E3))=
2
3
⋅
2
5
⋅
2
7
The probability that Q hits and both P and R miss is: P(E1∩E2∩E3)=(1−P(E1))⋅P(E2)⋅(1−P(E3))=
1
3
⋅
3
5
⋅
2
7
The probability that both P and Q hit but R misses is: P(E1∩E2∩E3)=P(E1)⋅P(E2)⋅(1−P(E3))=
2
3
⋅
3
5
⋅
2
7
Now, summing these probabilities gives the required probability that the target is hit by P or Q, but not by R : P( hit by P or Q but not R)=P(E1∩E2∩E3)+P(E1∩E2∩E3)+P(E1∩E2∩E3) =
2
3
⋅
2
5
⋅
2
7
+
1
3
⋅
3
5
⋅
2
7
+
2
3
⋅
3
5
⋅
2
7
=
8
105
+
6
105
+
12
105
=
26
105
Thus, the probability that the target is hit by P or Q but not by R is