Question

Assertion (A) : The probability that a leap year has 53 Mondays is \(\frac{2}{7}\).
Reason (R) : The probability that a non-leap year has 53 Mondays is \(\frac{5}{7}\).

• A. Both (A) and (R) are true and (R) is the correct explanation of (A).
• B. Both A and R are true but R is not the correct explanation of A
• C. A is true but R is false
• D. A is false but R is true.

Hint:

For any day of the week, the probability of it appearing 53 times is always \(1/7\) in a non-leap year and \(2/7\) in a leap year.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
A standard year has 365 days, and a leap year has 366 days. Since \(364 = 52 \times 7\), every year has at least 52 full weeks. The extra days determine the probability of having a 53rd occurrence of a specific weekday.
Step 2: Key Formula or Approach:
1. Non-leap year: 365 days = 52 weeks + 1 extra day. 2. Leap year: 366 days = 52 weeks + 2 extra days.
Step 3: Detailed Explanation:
1. Evaluating Assertion (A): In a leap year, there are 2 extra days. These could be (Mon, Tue), (Tue, Wed), (Wed, Thu), (Thu, Fri), (Fri, Sat), (Sat, Sun), or (Sun, Mon). Out of 7 possibilities, 2 contain a Monday. Thus, \(P(53 \text{ Mondays}) = 2/7\). Assertion (A) is true. 2. Evaluating Reason (R): In a non-leap year, there is only 1 extra day. It can be any of the 7 days of the week. The probability that it is a Monday is \(1/7\). Reason (R) states it is \(5/7\), which is false.
Step 4: Final Answer:
Assertion (A) is true, but Reason (R) is false.