Key Concepts and Formulas

To solve this problem, we will utilize fundamental concepts from probability theory, which are essential for JEE.

1. Probability of an Event and Complement Rule: The probability of an event $A$, denoted $P(A)$, measures its likelihood. The probability that an event $A$ does not occur, denoted $P(A')$, is given by: $P(A') = 1 - P(A)$ This rule is crucial for calculating the chance of an event's non-occurrence or for "at least one" scenarios.

2. Independence of Events: Events are independent if the outcome of one does not influence the outcome of the others. For independent events $A$ and $B$, the probability that both $A$ and $B$ occur is the product of their individual probabilities: $P(A \cap B) = P(A) \times P(B)$ A critical property is that if events $A$ and $B$ are independent, then their complements ($A'$ and $B'$) are also independent. This principle extends to any number of independent events.

3. Probability of "A or B" (Union of Events): The probability that event $A$ occurs or event $B$ occurs (or both), denoted $P(A \cup B)$, can be found using the inclusion-exclusion principle: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. For independent events $A$ and $B$, this simplifies to $P(A \cup B) = P(A) + P(B) - P(A)P(B)$. An alternative, and often simpler, approach for independent events is to use the complement rule: $P(A \cup B) = 1 - P(A' \cap B')$ Since $A$ and $B$ are independent, $A'$ and $B'$ are also independent, so $P(A' \cap B') = P(A')P(B')$. Thus, $P(A \cup B) = 1 - P(A')P(B')$ This formula is particularly efficient when calculating the probability of "at least one" success.

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Step-by-Step Solution

1. Define Events and Given Probabilities

First, let's clearly define the events and write down their given probabilities. This helps in organizing our thoughts and calculations.

• Let $P_H$ be the event that person P hits the target.
• Let $Q_H$ be the event that person Q hits the target.
• Let $R_H$ be the event that person R hits the target.

The problem provides the probabilities of each person hitting the target:

• $P(P_H) = \frac{3}{4}$
• $P(Q_H) = \frac{1}{2}$
• $P(R_H) = \frac{5}{8}$

2. Calculate Probabilities of Not Hitting the Target

Since we are interested in scenarios where people might not hit the target (e.g., P or Q hits, but R does not), we need to calculate the probabilities of their respective complement events. We use the Complement Rule for this.

• For P not hitting the target ($P_H'$): $P(P_H') = 1 - P(P_H) = 1 - \frac{3}{4} = \frac{1}{4}$ Explanation: The probability that P fails to hit the target is $1$ minus the probability that P hits the target.

• For Q not hitting the target ($Q_H'$): $P(Q_H') = 1 - P(Q_H) = 1 - \frac{1}{2} = \frac{1}{2}$ Explanation: Similarly, the probability that Q fails to hit the target is $1$ minus the probability that Q hits the target.

• For R not hitting the target ($R_H'$): $P(R_H') = 1 - P(R_H) = 1 - \frac{5}{8} = \frac{3}{8}$ Explanation: The probability that R fails to hit the target is $1$ minus the probability that R hits the target.

3. Interpret the Required Event and Break it Down

The problem asks for the probability that "the target is hit by P or Q but not by R". Let's break this complex event into simpler, independent parts:

• "Hit by P or Q": This means at least one of P or Q hits the target. This can be represented as the union of events $P_H \cup Q_H$.
• "Not by R": This means R does not hit the target. This is the event $R_H'$.

Since the attempts of P, Q, and R are stated to be independent, the event "(P or Q hits) AND (R does not hit)" can be treated as two independent events. Therefore, we can multiply their probabilities: $P(\text{(P or Q hits) AND (R does not hit)}) = P(P_H \cup Q_H) \times P(R_H')$ Explanation: The independence of P, Q, and R's attempts is crucial here. It allows us to multiply the probability of the outcome of P and Q's attempts by the probability of R's outcome.

4. Calculate the Probability of "P or Q Hitting the Target"

Now we need to find $P(P_H \cup Q_H)$. We use the formula for the union of two independent events: $P(A \cup B) = 1 - P(A')P(B')$. This is often simpler than using the inclusion-exclusion principle for "or" conditions involving independent events.

• Probability of neither P nor Q hitting the target ($P_H' \cap Q_H'$): Since P and Q are independent, their complements $P_H'$ and $Q_H'$ are also independent. $P(P_H' \cap Q_H') = P(P_H') \times P(Q_H') = \frac{1}{4} \times \frac{1}{2} = \frac{1}{8}$ Explanation: The event "neither P nor Q hits" means P fails and Q fails. Due to independence, we multiply their individual failure probabilities.

• Probability of P or Q hitting the target ($P_H \cup Q_H$): This is the complement of "neither P nor Q hitting the target". $P(P_H \cup Q_H) = 1 - P(P_H' \cap Q_H') = 1 - \frac{1}{8} = \frac{7}{8}$ Explanation: The event "P or Q hits" is the opposite of "neither P nor Q hits". If there's a $\frac{1}{8}$ chance that both fail, then there's a $1 - \frac{1}{8}$ chance that at least one of them succeeds.

5. Calculate the Final Probability

Finally, we combine the probabilities calculated in steps 3 and 4. We need the probability that "(P or Q hits) AND (R does not hit)".

• We found $P(P_H \cup Q_H) = \frac{7}{8}$.
• We found $P(R_H') = \frac{3}{8}$.

Since these two events (P or Q hitting, and R not hitting) are independent: $P(\text{(P or Q hits) AND (R does not hit)}) = P(P_H \cup Q_H) \times P(R_H')$ $= \frac{7}{8} \times \frac{3}{8}$ $= \frac{21}{64}$ Explanation: We multiply the probability that at least one of P or Q hits the target by the probability that R does not hit the target. This is allowed because all individual attempts are independent.

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Common Mistakes & Tips

• Misinterpreting "OR": A common mistake is to simply add probabilities for "A or B" ($P(A) + P(B)$). Remember, this is only correct if the events are mutually exclusive (cannot happen at the same time). For independent events that are not mutually exclusive, use $P(A \cup B) = P(A) + P(B) - P(A)P(B)$ or, more efficiently, $1 - P(A')P(B')$.
• Forgetting Independence: Always confirm if events are independent before multiplying their probabilities. The problem statement usually specifies this ("independently try to hit").
• Calculation Errors: Be careful with fractions, especially when adding or subtracting them. Ensure common denominators when necessary.

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Summary

This problem effectively tests your understanding of fundamental probability concepts: the complement rule and the independence of events. We systematically broke down the complex probability statement into smaller, manageable parts. We first defined the events and their given probabilities, then calculated the probabilities of their complements. We then interpreted the "P or Q hits" condition as the complement of "neither P nor Q hits" and calculated its probability. Finally, we combined this with the probability of "R not hitting" using the independence of events to find the overall probability.

The final probability that the target is hit by P or Q but not by R is $\frac{21}{64}$.

The final answer is $\boxed{\frac{21}{64}}$, which corresponds to option (A).