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Step-by-Step Solution
Step 1: Identify the given probabilities
Let the probabilities of hitting the target be:
• $P(\text{P hits}) = \tfrac{3}{4}$, so $P(\text{P does not hit}) = 1 - \tfrac{3}{4}=\tfrac{1}{4}$.
• $P(\text{Q hits}) = \tfrac{1}{2}$, so $P(\text{Q does not hit}) = 1 - \tfrac{1}{2}=\tfrac{1}{2}$.
• $P(\text{R hits}) = \tfrac{5}{8}$, so $P(\text{R does not hit}) = 1 - \tfrac{5}{8}=\tfrac{3}{8}$.
Step 2: Express the required probability
We need the probability that the target is hit by P or Q but not by R. That means R must miss, while at least one of P or Q must succeed. Mathematically, this scenario can be broken into three parts:
P hits, Q does not hit, R does not hit
Q hits, P does not hit, R does not hit
Both P and Q hit, R does not hit
Step 3: Calculate the probabilities for each scenario
P hits, Q does not hit, R does not hit:
$ \Bigl(\tfrac{3}{4}\Bigr)\Bigl(\tfrac{1}{2}\Bigr)\Bigl(\tfrac{3}{8}\Bigr)
= \tfrac{3}{4} \times \tfrac{1}{2} \times \tfrac{3}{8}
= \tfrac{9}{64}. $
Q hits, P does not hit, R does not hit:
$ \Bigl(\tfrac{1}{4}\Bigr)\Bigl(\tfrac{1}{2}\Bigr)\Bigl(\tfrac{3}{8}\Bigr)
= \tfrac{1}{4} \times \tfrac{1}{2} \times \tfrac{3}{8}
= \tfrac{3}{64}. $
Both P and Q hit, R does not hit:
$ \Bigl(\tfrac{3}{4}\Bigr)\Bigl(\tfrac{1}{2}\Bigr)\Bigl(\tfrac{3}{8}\Bigr)
= \tfrac{3}{4} \times \tfrac{1}{2} \times \tfrac{3}{8}
= \tfrac{9}{64}. $
Step 4: Sum the probabilities
The total probability that P or Q hits but not R is the sum of these three mutually exclusive events:
$ \tfrac{9}{64} + \tfrac{3}{64} + \tfrac{9}{64}
= \tfrac{21}{64}. $
Final Answer
The probability that the target is hit by P or Q but not by R is
$ \tfrac{21}{64}. $
