Problem 35 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

A, B, C try to hit a target simultaneously but independently. Their respective probabilities of hitting the targets are \(\frac{3}{4}, \frac{1}{2}, \frac{5}{8} .\) The probability that the target is hit by A or B but not by \(\mathrm{C}\) is : [Online April 23, 2013] (a) \(21 / 64\) (b) \(7 / 8\) (c) \(7 / 32\) (d) \(9 / 64\)

Step-by-Step Solution

Answer

The probability is \(\frac{9}{32}\), but it's not an option, re-check problem context.

Step 1: Understand the Probabilities

First, let's understand the given information: The probabilities for A, B, and C to hit the target are \( \frac{3}{4} \), \( \frac{1}{2} \), and \( \frac{5}{8} \) respectively. We need to find the probability that A or B hits the target, but C does not.

Step 2: Determine Probability That C Misses

Calculate the probability that C does not hit the target. This is given by \(1 - \text{Probability that C hits the target}\). Thus, \(1 - \frac{5}{8} = \frac{3}{8}\).

Step 3: Calculate A or B Hitting the Target

The event we want is when either A or B hits the target. The probability that A hits is \(\frac{3}{4}\) and B hits is \(\frac{1}{2}\). Since they are independent, use the formula for the probability of A or B hitting: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\).

Step 4: Calculate A and B Hitting the Target Together

Determine \(P(A \cap B)\), which is the probability that both A and B hit the target. For independent events, it is \(\frac{3}{4} \times \frac{1}{2} = \frac{3}{8}\).

Step 5: Calculate Probability that A or B Hits

Use the results from step 3 and 4: \[P(A \cup B) = \frac{3}{4} + \frac{1}{2} - \frac{3}{8} = \frac{9}{8} - \frac{3}{8} = \frac{6}{8} = \frac{3}{4}.\]

Step 6: Combine A or B Hitting With C Not Hitting

Finally, calculate the probability that A or B hits AND C does not hit. Use the formula \(P(A \cup B) \times P(C')\), where \(C'\) is the event C misses: \(\frac{3}{4} \times \frac{3}{8} = \frac{9}{32}.\)

Key Concepts

Independent Events

In probability, independent events are situations where the occurrence or non-occurrence of one event does not affect the outcome of another. When we consider A, B, and C trying to hit a target, each of their chances are independent. That means whether A hits doesn't change B's or C's chances to hit.
To recognize independent events, look for phrases indicating no influence between events, such as “simultaneously but independently.”

• If A has a probability \( \frac{3}{4} \) to hit the target, it stays the same irrespective of what B and C do.
• This enables us to use the multiplication rule for these scenarios. If two events, say A and B, are independent, their joint probability is found by multiplying their respective individual probabilities. \( P(A \cap B) = P(A) \times P(B) \).

Probability Theory

Probability theory is a branch of mathematics that deals with the analysis of random phenomena. It's useful in predicting outcomes of random events like hitting a target. The theory provides tools to calculate chances, making it a powerful tool to handle uncertainty.
One key principle is the concept of complementary events. An event and its complement cover all outcomes of an experiment. For example, if C has a probability of \( \frac{5}{8} \) to hit, then the complement, the event that C misses, has a probability of \( 1 - \frac{5}{8} = \frac{3}{8} \). This concept helps in calculating probabilities like in complex event scenarios.
Another key element in probability theory is the formula for the union of two events. If you want the probability of either A or B hitting the target, use \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]. This is crucial for cases involving "or" scenarios, ensuring we don't count any overlapping outcomes twice.

Probability Calculation

Once you understand the rules and concepts, calculating probability becomes methodical. Let's say you need to find the probability of a more specific event: A or B hits the target, but C does not. Here's a step-by-step way to approach this.
1. **Determine Individual Chances:** Confirm the probability for each event happening in isolation:

• A hits: \( \frac{3}{4} \)
• B hits: \( \frac{1}{2} \)
• C does not hit: Complement to C hitting \( \frac{3}{8} \)

2. **Apply Probability Formulas:** Use the overlapping scenario formula to find A or B hitting: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \ = \frac{3}{4} + \frac{1}{2} - \frac{3}{8} = \frac{3}{4} \]. 3. **Combine Events for Specific Outcomes:** Multiply the results by the complementary event for C: \[ P((A \cup B) \cap C') = P(A \cup B) \times P(C') = \frac{3}{4} \times \frac{3}{8} = \frac{9}{32} \].
This calculation framework ensures you're aligning with the rules of probability, setting you up to clearly see how each part contributes to the final result.