In a leap year with 366 days, what is the probability that the calendar contains exactly 53 Saturdays and exactly 52 Sundays?

Introduction / Context:
This question uses basic calendar facts combined with simple probability. A leap year has 366 days, which is 52 complete weeks plus 2 extra days. Every weekday must appear either 52 or 53 times. We are asked to find the probability that there are 53 Saturdays and only 52 Sundays in such a year. The problem reduces to understanding how the two extra days are distributed among the seven days of the week.

Given Data / Assumptions:

A leap year has 366 days.

366 days = 52 weeks + 2 days.

Each day of the week occurs at least 52 times.

Exactly two days of the week will occur 53 times.

The first day of the year (1 January) is equally likely to be any of the 7 weekdays.

Concept / Approach:
If we write out the calendar, the 52 full weeks contribute 52 of each weekday. The remaining 2 days decide which weekdays get a 53rd occurrence. If 1 January is some weekday, the two extra days are that weekday and the next one in order. Therefore, the pair of extra days can be (Monday, Tuesday), (Tuesday, Wednesday), and so on, cycling through the week. We check which choices of starting day give exactly 53 Saturdays and only 52 Sundays.

Step-by-Step Solution:
Step 1: In a leap year, there are 52 Saturdays and 52 Sundays guaranteed from the 52 complete weeks.Step 2: There are 2 extra days, which are consecutive weekdays.Step 3: List the possible ordered pairs of these extra days depending on the starting day: (Mon, Tue), (Tue, Wed), (Wed, Thu), (Thu, Fri), (Fri, Sat), (Sat, Sun), (Sun, Mon).Step 4: To have 53 Saturdays, Saturday must appear among these extra days.Step 5: To have exactly 52 Sundays, Sunday must not appear among these extra days.Step 6: Among the listed pairs, the pair (Fri, Sat) includes Saturday but not Sunday. The pair (Sat, Sun) includes both Saturday and Sunday, which would give 53 Saturdays and 53 Sundays, not allowed.Step 7: Therefore, the only favourable case is when the extra days are (Fri, Sat), which occurs when 1 January is a Friday.Step 8: There are 7 equally likely choices for the day on which 1 January falls, and exactly 1 of these choices is favourable.Step 9: Hence, probability = 1 / 7.

Verification / Alternative check:
We can verify by reasoning directly on Saturdays. There are two ways for Saturday to be an extra day: the year starts on Friday or Saturday. However, when it starts on Saturday, the second extra day is Sunday, giving 53 Sundays as well. Since we require only 52 Sundays, only the Friday start works, reinforcing the answer 1 / 7.

Why Other Options Are Wrong:
2/7 and 3/7 treat multiple starting days as favourable, including cases where Sunday also reaches 53 occurrences. The value 1/2 suggests that roughly half of all leap years satisfy the condition, which is far too large. The value 4/7 similarly overcounts many starts without checking the Sunday condition.

Common Pitfalls:
Learners sometimes stop as soon as they find all starts that give 53 Saturdays and forget to enforce the requirement that Sundays must stay at 52. Others misremember whether leap years add one day or two days beyond full weeks. Always remember that a leap year gives 52 full weeks and 2 extra days, and carefully check both conditions in probability questions involving days of the week.

Final Answer:
The probability that a leap year has exactly 53 Saturdays and 52 Sundays is 1/7.