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In Mathematics / College | 2025-07-04

A leap year is selected at random. What is the probability that it will contain 53 Sundays?
a) $\frac{1}{7}$
b) $\frac{2}{7}$
c) $\frac{53}{7}$
d) $\frac{53}{365}$

Asked by acharyagonin34

Answer (1)

A leap year has 366 days, which is 52 weeks and 2 days.
The 2 extra days can be any of the following pairs: (Saturday, Sunday), (Sunday, Monday), (Monday, Tuesday), (Tuesday, Wednesday), (Wednesday, Thursday), (Thursday, Friday), (Friday, Saturday).
For the leap year to have 53 Sundays, one of these two extra days must be a Sunday. There are 2 such pairs: (Saturday, Sunday) and (Sunday, Monday).
The probability is 7 2 ​ ​ .

Explanation

Understand the problem and provided data A leap year has 366 days. We want to find the probability that a randomly selected leap year has 53 Sundays.

Calculate the number of weeks and extra days First, let's determine how many weeks and extra days are in a leap year. Since a leap year has 366 days, we divide 366 by 7 to find the number of weeks and remaining days.

Determine the number of full weeks The number of full weeks is ⌊ 7 366 ​ ⌋ = 52 .

Determine the number of extra days The number of extra days is 366 mod 7 = 2 .

List all possible pairs of extra days These 2 extra days can be any of the following pairs: (Saturday, Sunday), (Sunday, Monday), (Monday, Tuesday), (Tuesday, Wednesday), (Wednesday, Thursday), (Thursday, Friday), (Friday, Saturday).

Identify the condition for 53 Sundays For the leap year to have 53 Sundays, one of these two extra days must be a Sunday.

Count the favorable outcomes Count the number of pairs that contain a Sunday. There are 2 such pairs: (Saturday, Sunday) and (Sunday, Monday).

Calculate the probability Calculate the probability by dividing the number of favorable outcomes (pairs with a Sunday) by the total number of possible outcomes (all possible pairs). The probability is 7 2 ​ .


Examples
Consider a game where you randomly pick a year and bet on whether it has 53 Sundays. Knowing the probability for a leap year helps you assess your odds and make informed betting decisions. This is also useful in scheduling events, ensuring that you account for the likelihood of having an extra Sunday in certain years.

Answered by GinnyAnswer | 2025-07-04

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