Key Concepts and Formulas

This problem requires us to determine the probability that a target is hit when multiple independent events (persons hitting the target) occur. The phrase "the target would be hit" implies that at least one of the persons hits the target. To tackle such "at least one" scenarios efficiently, we primarily use two fundamental probability concepts:

1. Probability of Complementary Events (The Complement Rule): For any event $E$, the probability that $E$ occurs, $P(E)$, is related to the probability that $E$ does not occur (its complement, denoted as $\bar{E}$ or $E^c$) by the formula: $P(E) = 1 - P(\bar{E})$ This rule is particularly useful when calculating the probability of "at least one" event, as it's often simpler to calculate the probability of "none of the events" occurring and subtract it from 1.

2. Probability of Independent Events: If two or more events are independent, the probability that all of them occur simultaneously is the product of their individual probabilities. For independent events $E_1, E_2, \ldots, E_n$: $P(E_1 \cap E_2 \cap \ldots \cap E_n) = P(E_1) \cdot P(E_2) \cdot \ldots \cdot P(E_n)$ The problem explicitly states that all persons hit the target "independently," meaning one person's outcome does not influence another's. This allows us to multiply probabilities directly.

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Step-by-Step Solution

Step 1: Define Events and List Given Probabilities

First, let's clearly define the events for each person hitting the target and list their given probabilities.

• Let $A$ be the event that the first person hits the target.
• Let $B$ be the event that the second person hits the target.
• Let $C$ be the event that the third person hits the target.
• Let $D$ be the event that the fourth person hits the target.

The given probabilities of hitting the target are:

• $P(A) = \frac{1}{2}$
• $P(B) = \frac{1}{3}$
• $P(C) = \frac{1}{4}$
• $P(D) = \frac{1}{8}$

Why this step? Defining events and listing probabilities provides a clear foundation for our calculations, making the problem easier to follow and ensuring accuracy.

Step 2: Calculate Probabilities of Complementary Events (Not Hitting the Target)

Since our strategy involves using the complement rule for the overall problem ("at least one hit"), we first need to find the probability that each person fails to hit the target. Let $\bar{A}$ denote the event that person A does not hit the target, and similarly for $\bar{B}$, $\bar{C}$, and $\bar{D}$.

Using the complement rule $P(\bar{E}) = 1 - P(E)$:

• $P(\bar{A}) = 1 - P(A) = 1 - \frac{1}{2} = \frac{1}{2}$
• $P(\bar{B}) = 1 - P(B) = 1 - \frac{1}{3} = \frac{2}{3}$
• $P(\bar{C}) = 1 - P(C) = 1 - \frac{1}{4} = \frac{3}{4}$
• $P(\bar{D}) = 1 - P(D) = 1 - \frac{1}{8} = \frac{7}{8}$

Why this step? To find the probability that the target is not hit by anyone, we need the individual probabilities of each person failing to hit the target. These are the building blocks for the complementary event.

Step 3: Formulate the Required Probability using the Complement Rule

Let $H$ be the event that the target is hit. This means $P(H)$ is the probability that at least one of the persons A, B, C, or D hits the target. As discussed in the "Key Concepts," it's significantly simpler to find the probability that the target is not hit ($\bar{H}$), and then subtract that from 1. The event $\bar{H}$ (target is not hit) occurs if and only if none of the four persons hit the target. This means person A does not hit AND person B does not hit AND person C does not hit AND person D does not hit. In terms of events, this is the intersection of their complement events: $\bar{A} \cap \bar{B} \cap \bar{C} \cap \bar{D}$.

So, we can express the desired probability as: $P(H) = 1 - P(\bar{H}) = 1 - P(\bar{A} \cap \bar{B} \cap \bar{C} \cap \bar{D})$

Why this step? This is the crucial strategic decision. By converting "at least one hit" into "1 - none hit", we simplify a potentially complex sum of probabilities (1 hit, 2 hits, 3 hits, 4 hits) into a single, more manageable calculation.

Step 4: Apply the Property of Independent Events to Calculate $P(\bar{H})$

Since the events of each person hitting the target are independent, it logically follows that the events of each person not hitting the target are also independent. Therefore, the probability that none of them hit the target is the product of their individual probabilities of not hitting the target: $P(\bar{H}) = P(\bar{A} \cap \bar{B} \cap \bar{C} \cap \bar{D}) = P(\bar{A}) \cdot P(\bar{B}) \cdot P(\bar{C}) \cdot P(\bar{D})$

Why this step? The independence property is key here. It allows us to simply multiply the individual failure probabilities together, which is a direct application of the definition of independent events. Without independence, this multiplication would not be valid.

Step 5: Substitute Values and Perform Intermediate Calculation

Now, we substitute the probabilities of not hitting the target (calculated in Step 2) into the equation from Step 4: $P(\text{none hit the target}) = \left(\frac{1}{2}\right) \cdot \left(\frac{2}{3}\right) \cdot \left(\frac{3}{4}\right) \cdot \left(\frac{7}{8}\right)$ To simplify this product, we can cancel common terms in the numerator and denominator: $P(\text{none hit the target}) = \frac{1}{\cancel{2}} \cdot \frac{\cancel{2}}{\cancel{3}} \cdot \frac{\cancel{3}}{4} \cdot \frac{7}{8}$ $P(\text{none hit the target}) = \frac{1 \cdot 1 \cdot 1 \cdot 7}{1 \cdot 1 \cdot 4 \cdot 8}$ $P(\text{none hit the target}) = \frac{7}{32}$ This is the probability that none of the persons hit the target.

Why this step? This calculation yields the probability of the complementary event, $P(\bar{H})$, which is a necessary intermediate step before applying the complement rule to find the final answer. Careful fraction multiplication and simplification prevent errors.

Step 6: Final Calculation using the Complement Rule

Finally, we use the result from Step 5 and the formula from Step 3 to calculate the probability that the target is hit: $P(H) = 1 - P(\text{none hit the target})$ $P(H) = 1 - \frac{7}{32}$ To perform the subtraction, we express 1 with the same denominator as the fraction: $P(H) = \frac{32}{32} - \frac{7}{32}$ $P(H) = \frac{32 - 7}{32}$ $P(H) = \frac{25}{32}$

Why this step? This is the final step that directly answers the question. By subtracting the probability of the complementary event from 1, we obtain the desired probability of the target being hit.

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Common Mistakes & Tips

• "At Least One" is a Strong Hint: Whenever a probability problem asks for the chance of "at least one" event occurring, immediately consider using the complement rule: $1 - P(\text{none of the events occur})$. This is almost always the most efficient path.
• Verify Independence: The multiplication rule for probabilities (e.g., $P(E_1 \cap E_2) = P(E_1) \cdot P(E_2)$) is valid only if the events are independent. Always confirm this condition is stated (as it is here) or can be reasonably assumed.
• Careful with Complement Probabilities: Ensure you correctly calculate $P(\bar{E}) = 1 - P(E)$. A frequent error is to accidentally use the original probability $P(E)$ when $P(\bar{E})$ is required, especially in multi-step problems.
• Fraction Arithmetic: Pay meticulous attention to fraction operations (addition, subtraction, multiplication). Look for opportunities to cancel common factors in numerators and denominators to simplify calculations and reduce potential errors.

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Summary

This problem is a classic application of the complement rule combined with the property of independent events. To find the probability that the target is hit (meaning at least one person hits it), we first calculated the probability that none of the four persons hit the target. This was done by finding each person's probability of not hitting the target and then multiplying these probabilities together, leveraging the independence of their shots. Finally, we subtracted this product from 1 to get the desired probability. This method efficiently avoids the more complex calculation of summing probabilities for 1, 2, 3, or 4 hits.

The final answer is $\boxed{\frac{25}{32}}$, which corresponds to option (A).