A leap year is selected at random. What is the probability that this leap year will contain exactly 53 Mondays?

Introduction / Context:
This question connects basic probability with calendar facts. It tests whether you know how many days are in a leap year and how weekdays repeat. The focus is on working out how often a specific day, Monday, occurs in a leap year and expressing that as a probability when the starting weekday is not fixed.

Given Data / Assumptions:

• A leap year has 366 days.
• 366 days is equal to 52 full weeks plus 2 extra days.
• Each day of the week is equally likely to be the first day of the leap year.
• We want the leap year to have exactly 53 Mondays.

Concept / Approach:
In any leap year, every day of the week occurs at least 52 times because 52 weeks cover 364 days. The remaining 2 days fall on two consecutive weekdays. Therefore, exactly two weekdays will occur 53 times. We need to count how many choices for the first day of the year make Monday one of the two weekdays that occur 53 times, and then divide by the total possible starting weekdays (7).

Step-by-Step Solution:
366 days = 52 weeks (364 days) + 2 extra days. Every weekday appears at least 52 times. The extra 2 days are consecutive weekdays starting from the weekday of 1 January. If 1 January is Monday, then Monday and Tuesday occur 53 times. If 1 January is Sunday, then Sunday and Monday occur 53 times. So Monday appears 53 times if 1 January is Monday or Sunday (2 cases). Total possible starting weekdays = 7. Required probability = 2 / 7.

Verification / Alternative check:
We can systematically consider all seven possible starting weekdays: Sunday to Saturday. For each, list the two weekdays that occur 53 times (starting day and next day). Only when the starting day is Sunday or Monday does Monday appear among these two. That again gives 2 favourable cases out of 7 possibilities.

Why Other Options Are Wrong:
1/7: This would mean only one starting weekday leads to 53 Mondays, which is not correct. 3/7: This implies three possible starting days, but we found only two. 1: This would mean every leap year has 53 Mondays, which is clearly false.

Common Pitfalls:
Some learners divide by 366 directly or guess without using the week structure. Others forget that in a leap year there are exactly two weekdays with 53 occurrences and the rest have 52. The right way is to track the starting weekday and see how the additional two days distribute.

Final Answer:
Therefore, the probability that a randomly chosen leap year has exactly 53 Mondays is 2/7.