Three persons P, Q and R independentlytry to hit a target. If the probabilities oftheir hitting the target are `3/4,1/2 and 5/8` respectively, then the probability that thetarget is hit by P or Q but not by R is:

Options

A. `(39)/(64)`
B. `(21)/(64)`
C. `(15)/(64)`
D. `(9)/(64)`

Answer

To solve the problem step-by-step, we need to find the probability that the target is hit by P or Q but not by R. Let's break it down:

### Step 1: Define the probabilities
- Let \( P \) be the probability that person P hits the target:
\[ P(P) = \frac{3}{4} \]
- Let \( Q \) be the probability that person Q hits the target:
\[ P(Q) = \frac{1}{2} \]
- Let \( R \) be the probability that person R hits the target:
\[ P(R) = \frac{5}{8} \]

### Step 2: Calculate the probabilities of not hitting the target
- The probability that P does not hit the target:
\[ P(\text{not } P) = 1 - P(P) = 1 - \frac{3}{4} = \frac{1}{4} \]
- The probability that Q does not hit the target:
\[ P(\text{not } Q) = 1 - P(Q) = 1 - \frac{1}{2} = \frac{1}{2} \]
- The probability that R does not hit the target:
\[ P(\text{not } R) = 1 - P(R) = 1 - \frac{5}{8} = \frac{3}{8} \]

### Step 3: Identify the cases where P or Q hits but not R
We need to consider three cases:
1. Both P and Q hit, and R does not hit.
2. Only P hits, and Q does not hit, and R does not hit.
3. Only Q hits, and P does not hit, and R does not hit.

### Step 4: Calculate the probabilities for each case
1. **Case 1**: Both P and Q hit, R does not hit:
\[ P_1 = P(P) \times P(Q) \times P(\text{not } R) = \frac{3}{4} \times \frac{1}{2} \times \frac{3}{8} \]
\[ P_1 = \frac{3 \times 1 \times 3}{4 \times 2 \times 8} = \frac{9}{64} \]

2. **Case 2**: Only P hits, Q does not hit, R does not hit:
\[ P_2 = P(P) \times P(\text{not } Q) \times P(\text{not } R) = \frac{3}{4} \times \frac{1}{2} \times \frac{3}{8} \]
\[ P_2 = \frac{3 \times 1 \times 3}{4 \times 2 \times 8} = \frac{9}{64} \]

3. **Case 3**: Only Q hits, P does not hit, R does not hit:
\[ P_3 = P(\text{not } P) \times P(Q) \times P(\text{not } R) = \frac{1}{4} \times \frac{1}{2} \times \frac{3}{8} \]
\[ P_3 = \frac{1 \times 1 \times 3}{4 \times 2 \times 8} = \frac{3}{64} \]

### Step 5: Calculate the total probability
Now, we sum the probabilities of all three cases:
\[ P(\text{total}) = P_1 + P_2 + P_3 = \frac{9}{64} + \frac{9}{64} + \frac{3}{64} = \frac{21}{64} \]

### Final Answer
The probability that the target is hit by P or Q but not by R is:
\[ \frac{21}{64} \]