A, B, C try to hit a target simultaneously but independently. Their respective probabilities of hitting the target are `3/4,1/2,5/8`. The probability that target is hit by A or B but not by C is

Options

A. `21/64`
B. `7/8`
C. `7/32`
D. `9/64`

Answer

To solve the problem, we need to find the probability that the target is hit by A or B but not by C. Let's break down the steps systematically.

### Step 1: Identify the probabilities
The probabilities of hitting the target are given as:
- Probability of A hitting the target, \( P(A) = \frac{3}{4} \)
- Probability of B hitting the target, \( P(B) = \frac{1}{2} \)
- Probability of C hitting the target, \( P(C) = \frac{5}{8} \)

### Step 2: Calculate the probabilities of not hitting the target
We need to find the probabilities that each person does not hit the target:
- Probability of A not hitting the target, \( P(A') = 1 - P(A) = 1 - \frac{3}{4} = \frac{1}{4} \)
- Probability of B not hitting the target, \( P(B') = 1 - P(B) = 1 - \frac{1}{2} = \frac{1}{2} \)
- Probability of C not hitting the target, \( P(C') = 1 - P(C) = 1 - \frac{5}{8} = \frac{3}{8} \)

### Step 3: Calculate the probability of A hitting and C not hitting
The probability that A hits the target and C does not hit the target is:
\[ P(A \text{ hits and } C' \text{ hits}) = P(A) \times P(C') = \frac{3}{4} \times \frac{3}{8} = \frac{9}{32} \]

### Step 4: Calculate the probability of B hitting and C not hitting
The probability that B hits the target and C does not hit the target is:
\[ P(B \text{ hits and } C' \text{ hits}) = P(B) \times P(C') = \frac{1}{2} \times \frac{3}{8} = \frac{3}{16} \]

### Step 5: Calculate the probability of A not hitting and B hitting and C not hitting
The probability that A does not hit, B hits, and C does not hit is:
\[ P(A' \text{ hits and } B \text{ hits and } C') = P(A') \times P(B) \times P(C') = \frac{1}{4} \times \frac{1}{2} \times \frac{3}{8} = \frac{3}{64} \]

### Step 6: Combine the probabilities
Now we need to combine the probabilities from steps 3, 4, and 5:
\[ P(A \text{ or } B \text{ hits but not } C) = P(A \text{ hits and } C') + P(B \text{ hits and } C') = \frac{9}{32} + \frac{3}{16} \]
To add these fractions, we need a common denominator. The least common multiple of 32 and 16 is 32:
\[ \frac{3}{16} = \frac{3 \times 2}{16 \times 2} = \frac{6}{32} \]
Now, we can add:
\[ P(A \text{ or } B \text{ hits but not } C) = \frac{9}{32} + \frac{6}{32} = \frac{15}{32} \]

### Final Answer
The probability that the target is hit by A or B but not by C is \( \frac{15}{32} \).

---