Probability

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 ‌

26

105

.