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

To solve the given problem, we need to verify the assertions about probabilities concerning the days in a leap year and a non-leap year.

1. Understanding a Leap Year:
   A leap year has 366 days. Dividing by 7 (the number of days in a week), we have: \(\frac{366}{7} = 52\) weeks and 2 days. This implies that a leap year will have 52 complete weeks plus 2 extra days.
2. Possible Combinations of Extra Days in a Leap Year:
   The extra two days can be any of the following combinations:
   • Sunday & Monday
   • Monday & Tuesday
   • Tuesday & Wednesday
   • Wednesday & Thursday
   • Thursday & Friday
   • Friday & Saturday
   • Saturday & Sunday
3. Probability of 53 Mondays in a Leap Year:
   For a leap year to have 53 Mondays, one of the two extra days must be a Monday. From the combinations listed, only "Sunday & Monday" and "Monday & Tuesday" include a Monday. Therefore, there are 2 favorable outcomes out of 7 possible combinations.
   
   The probability is \(\frac{2}{7}\).
4. Understanding a Non-Leap Year:
   A non-leap year has 365 days. Dividing by 7, we have: \(\frac{365}{7} = 52\) weeks and 1 day. This means a non-leap year will have 52 complete weeks plus 1 extra day.
5. Probability of 53 Mondays in a Non-Leap Year:
   The extra day can be any day of the week (Monday through Sunday). Thus, there are 7 possible outcomes for the extra day.
   
   Only 1 of these days is a Monday. Therefore, the probability is \(\frac{1}{7}\), not \(\frac{5}{7}\) as stated in the reason.

Conclusion:
From the analysis, the assertion (A) is true as the probability that a leap year has 53 Mondays is indeed \(\frac{2}{7}\). However, the reasoning (R) is false because the correct probability for a non-leap year having 53 Mondays is \(\frac{1}{7}\), not \(\frac{5}{7}\). Therefore, the correct answer is: A is true but R is false.