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 $\frac{5}{7}$.
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 $\frac{5}{7}$.
Reason (R) : The probability that a non-leap year has 53 Mondays is $\frac{5}{7}$.
Show Hint
For any day of the week (Monday, Tuesday, etc.):
- Probability of 53 occurrences in a leap year is always $\frac{2}{7}$.
- Probability of 53 occurrences in a non-leap year is always $\frac{1}{7}$.
Memorizing these two standard fractions will save you from drawing out sample spaces in exams!
- Probability of 53 occurrences in a leap year is always $\frac{2}{7}$.
- Probability of 53 occurrences in a non-leap year is always $\frac{1}{7}$.
Memorizing these two standard fractions will save you from drawing out sample spaces in exams!
Updated On: Jul 9, 2026
- Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
- Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
- Assertion (A) is true, but Reason (R) is false.
- Assertion (A) is false, but Reason (R) is true.
Show Solution
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The Correct Option is C
Solution and Explanation
Step 1: Understanding the Question:
This is an Assertion-Reason question assessing the probability of calendar events.
We need to analyze the probability of having 53 Mondays in:
1. A leap year (Assertion A).
2. A non-leap year (Reason R).
Step 2: Key Formula or Approach:
- A non-leap year has 365 days.
- A leap year has 366 days.
- One week has 7 days. We can find the number of complete weeks and remaining extra days (leap/non-leap) to calculate probabilities.
Step 3: Detailed Explanation:
• Evaluate Assertion (A):
A leap year contains 366 days.
Let us divide 366 by 7 to find the number of complete weeks:
\[ 366 = 52 \times 7 + 2 \]
This means a leap year has 52 complete weeks and 2 extra days.
The 52 complete weeks guarantee exactly 52 Mondays.
The remaining 2 consecutive extra days can be any of the following 7 pairs:
- (Sunday, Monday)
- (Monday, Tuesday)
- (Tuesday, Wednesday)
- (Wednesday, Thursday)
- (Thursday, Friday)
- (Friday, Saturday)
- (Saturday, Sunday)
Out of these 7 possible outcomes, the outcomes containing a Monday are:
- (Sunday, Monday)
- (Monday, Tuesday)
Thus, there are 2 favorable outcomes.
\[ P(53 \text{ Mondays in a leap year}) = \frac{2}{7} \]
So, Assertion (A) is true.
• Evaluate Reason (R):
A non-leap year contains 365 days.
Divide 365 by 7 to find the number of complete weeks:
\[ 365 = 52 \times 7 + 1 \]
This means a non-leap year has 52 complete weeks and 1 extra day.
The 52 complete weeks guarantee 52 Mondays.
The 1 extra day can be any of the 7 days of the week:
\[ \{\text{Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}\} \]
The favorable outcome for a 53rd Monday is that the extra day is a Monday (1 favorable outcome).
Therefore, the probability of having 53 Mondays in a non-leap year is:
\[ P(53 \text{ Mondays in a non-leap year}) = \frac{1}{7} \]
The Reason statement claims this probability is $\frac{5}{7}$, which is incorrect.
Thus, Reason (R) is false.
Step 4: Final Answer:
Assertion (A) is true, but Reason (R) is false.
Hence, option (C) is correct.
This is an Assertion-Reason question assessing the probability of calendar events.
We need to analyze the probability of having 53 Mondays in:
1. A leap year (Assertion A).
2. A non-leap year (Reason R).
Step 2: Key Formula or Approach:
- A non-leap year has 365 days.
- A leap year has 366 days.
- One week has 7 days. We can find the number of complete weeks and remaining extra days (leap/non-leap) to calculate probabilities.
Step 3: Detailed Explanation:
• Evaluate Assertion (A):
A leap year contains 366 days.
Let us divide 366 by 7 to find the number of complete weeks:
\[ 366 = 52 \times 7 + 2 \]
This means a leap year has 52 complete weeks and 2 extra days.
The 52 complete weeks guarantee exactly 52 Mondays.
The remaining 2 consecutive extra days can be any of the following 7 pairs:
- (Sunday, Monday)
- (Monday, Tuesday)
- (Tuesday, Wednesday)
- (Wednesday, Thursday)
- (Thursday, Friday)
- (Friday, Saturday)
- (Saturday, Sunday)
Out of these 7 possible outcomes, the outcomes containing a Monday are:
- (Sunday, Monday)
- (Monday, Tuesday)
Thus, there are 2 favorable outcomes.
\[ P(53 \text{ Mondays in a leap year}) = \frac{2}{7} \]
So, Assertion (A) is true.
• Evaluate Reason (R):
A non-leap year contains 365 days.
Divide 365 by 7 to find the number of complete weeks:
\[ 365 = 52 \times 7 + 1 \]
This means a non-leap year has 52 complete weeks and 1 extra day.
The 52 complete weeks guarantee 52 Mondays.
The 1 extra day can be any of the 7 days of the week:
\[ \{\text{Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}\} \]
The favorable outcome for a 53rd Monday is that the extra day is a Monday (1 favorable outcome).
Therefore, the probability of having 53 Mondays in a non-leap year is:
\[ P(53 \text{ Mondays in a non-leap year}) = \frac{1}{7} \]
The Reason statement claims this probability is $\frac{5}{7}$, which is incorrect.
Thus, Reason (R) is false.
Step 4: Final Answer:
Assertion (A) is true, but Reason (R) is false.
Hence, option (C) is correct.
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