Question

The probability that a leap year shoold have 53 Sundays is ...

Short answer

\( \frac{2}{7} \)

Step-by-step solution

Determine the total days in a leap year

A leap year contains 366 days. If you divide these days by 7, which represent the days of the week, you will find that there are 52 full weeks in a leap year and 2 extra days.

Identify all possible combinations of the two extra days

The extra days can take the following combinations of days: (Sunday, Monday), (Monday, Tuesday), (Tuesday, Wednesday), (Wednesday, Thursday), (Thursday, Friday), (Friday, Saturday), or (Saturday, Sunday).

Determine when there is an extra Sunday

Out of the 7 combinations, two have an extra Sunday: (Sunday, Monday) and (Saturday, Sunday).

Calculate the probability

Since there are two favorable outcomes for an extra Sunday and 7 possible combinations, the probability that a leap year will have 53 Sundays is given by \( \frac{2}{7} \).

Key concepts

Leap Year

A leap year is a special year that occurs every four years. It's unique because it contains 366 days instead of the usual 365. This extra day is added to the month of February, which becomes 29 days instead of 28.
The reason we have leap years is because a complete revolution around the sun takes approximately 365.25 days. To make our calendar year consistent with the astronomical year, we add an extra day every four years, thereby compensating for the quarter day gained each year.

• Ordinary year: 365 days
• Leap year: 366 days

Keeping track of this ensures that our seasons and calendar dates remain aligned with Earth's position in its orbit around the sun.

Days of the Week

Days of the week are the repeating cycle of seven days that make up our weekly structure. They are named as follows: Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, and Saturday.
This weekly pattern arises from historical and cultural conventions that are centuries old. In the context of a leap year, understanding the days of the week is crucial to solving problems related to 53 Sundays, or any other day.
When a leap year, with its 366 days, is divided by the 7 days in a week, we find 52 full weeks plus 2 extra days.

• 52 weeks x 7 days = 364 days
• 366 - 364 = 2 extra days

These two extra days are what affect the frequency of the occurrence of specific days like Sundays.

Combinatorics

Combinatorics is a branch of mathematics focused on counting and arranging possibilities. In simple terms, it helps us figure out how many ways there are to do or arrange something.
In the case of a leap year having 53 Sundays, combinatorics helps in determining all possible combinations of the additional two days left after accounting for the 52 weeks.
The two extra days can form different pairs, such as (Sunday, Monday) and so on, leading to a total of 7 possible combinations.
Using combinatorics:

• Total combinations of extra days: 7
• Favorable combinations including Sunday: 2

This mathematical process helps us compute the probability that one of these combinations results in an extra Sunday.

Week Structure

Our week structure consists of seven days, a concept rooted deeply in various historical and cultural traditions. This repetitive cycle plays a significant role in calendar calculations and probability problems, such as determining the occurrence of 53 Sundays in a year.
The structure of the week doesn't change, providing a stable basis for calculations and planning.

• Each week: 7 days
• Total weeks in a leap year: 52 weeks plus 2 additional days

The extra days in a leap year can shift the occurrence of days throughout the year, influencing which day appears 53 times. The regularity of 7-day weeks across cultures makes identifying patterns easier in mathematical problems involving time.