A leap year has 366 days.\nIf a leap year is selected at random, what is the probability that it contains exactly 53 Sundays?

Introduction / Context:
This question involves calendar-based probability. You need to know how the days of the week distribute in a leap year and then determine the chance that there are exactly 53 Sundays. Understanding how extra days beyond full weeks work is the key idea.

Given Data / Assumptions:

• A leap year has 366 days.
• 366 days equal 52 full weeks plus 2 extra days.
• Each year can start on any day of the week with equal likelihood.
• We want the probability that there are 53 Sundays in the leap year.

Concept / Approach:
In any year, each day of the week occurs at least 52 times. In a leap year there are 2 extra days beyond the 52 full weeks. These 2 extra days determine which days of the week occur 53 times. There are 7 possible ordered pairs of consecutive extra days, depending on the starting day of the year.

Step-by-Step Solution:
Total days in a leap year = 366 = 52 weeks + 2 days. Every day of the week appears at least 52 times. The 2 extra days are consecutive days of the week. Possible ordered pairs of extra days: (Sun, Mon), (Mon, Tue), (Tue, Wed), (Wed, Thu), (Thu, Fri), (Fri, Sat), (Sat, Sun). There are 7 equally likely possibilities. A leap year will have 53 Sundays if Sunday is one of the extra two days. This happens when the pair is (Sun, Mon) or (Sat, Sun), so there are 2 favourable cases.

Verification / Alternative check:
We can consider the starting day. If the year starts on Sunday, the extra days are Sunday and Monday, giving 53 Sundays. If it starts on Saturday, the extra days are Saturday and Sunday, again giving 53 Sundays. For any other starting day, Sunday is not among the extra two, so there are only 52 Sundays. Therefore the probability is 2 favourable starting days out of 7.

Why Other Options Are Wrong:
Option 1/7 corresponds to only one favourable case, which is not correct. Option 1/3 is approximately 0.333 but the correct probability is about 0.2857. Option 4/7 is greater than 0.5 and would require four favourable starting days, which does not match the actual possibilities.

Common Pitfalls:
A frequent error is to miscount the number of extra days in a leap year or to assume each day appears the same number of times. Another mistake is to think that any year has 53 Sundays with probability 1/7, which is not accurate for leap years with two extra days. Always consider how many extra days the year has and how they are distributed.

Final Answer:
The probability that a leap year has exactly 53 Sundays is 2/7.