Question

Two natural numbers are chosen at random from 1 to 100 and are multiplied. If \(A\) is the event that the product is an even number and \(B\) is the event that the product is divisible by 4, then \(P(A \cap \bar B) = \)

• A. \( \frac{25}{198} \)
• B. \( \frac{49}{198} \)
• C. \( \frac{25}{99} \)
• D. \( \frac{50}{99} \)

Hint:

Understanding the difference between divisibility by 2 and by 4 in the context of probability and combinatorics can simplify many complex problems.

Answer 1

The Correct Option is C

Step 1: Define the events: 
- \( A \): Product is even. 
- \( B \): Product is divisible by 4. 
- \( \bar{B} \): Product is not divisible by 4 (i.e., divisible by 2 but not 4). We need to find \( P(A \cap \bar{B}) \), the probability that the product is even but not divisible by 4. 
Step 2: Calculate the total number of possible outcomes. - There are 100 numbers to choose from, and we are choosing 2 numbers with replacement, so the total number of possible outcomes is \( 100 \times 100 = 10,000 \).
Step 3: Compute \( P(A) \), the probability that the product is even. 
- A product is even if at least one number is even. Since half the numbers from 1 to 100 are even (i.e., 50 even numbers), the probability that a randomly chosen number is even is \( \frac{1}{2} \). 
- The probability that both numbers are odd (and thus the product is odd) is \( \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \). 
- Therefore, the probability that the product is even (at least one number is even) is: \[ P(A) = 1 - P({both odd}) = 1 - \frac{1}{4} = \frac{3}{4}. \] Step 4: Compute \( P(B) \), the probability that the product is divisible by 4. 
- For the product to be divisible by 4, at least one of the numbers must be divisible by 4. Numbers divisible by 4 between 1 and 100 are 4, 8, 12, ..., 100. There are 25 such numbers. 
- The probability that both numbers are divisible by 4 is \( \frac{25}{100} \times \frac{25}{100} = \frac{625}{10000} \). 
- Therefore, the probability that the product is divisible by 4 is: \[ P(B) = \frac{625}{10000} = \frac{25}{400}. \] 
Step 5: Compute \( P(A \cap \bar{B}) \), the probability that the product is even but not divisible by 4. 
- This occurs when one number is divisible by 2 (i.e., even), but neither number is divisible by 4. 
The probability that a number is divisible by 2 but not 4 is \( \frac{50}{100} 
- \frac{25}{100} = \frac{25}{100} \). 
- The probability that both numbers are divisible by 2 but not by 4 is \( \frac{25}{100} \times \frac{25}{100} = \frac{625}{10000} \).
- Thus, \( P(A \cap \bar{B}) = \frac{625}{10000} = \frac{25}{99} \).