Question

A year selected at random has 365 days. The probability that it has 53 Sundays is:

• A. \(\frac{1}{7}\)
• B. \(\frac{2}{7}\)
• C. \(\frac{1}{365}\)
• D. \(\frac{53}{365}\)

Hint:

Calendar probability depends on starting day.

Answer 1

The Correct Option is A

To determine the probability that a random year has 53 Sundays, let's analyze the situation:

1. Understanding the Structure of a Year: A typical year has 365 days. These 365 days can be divided into 52 weeks and 1 extra day (since 365 ÷ 7 = 52 weeks and 1 day).
2. Identifying the Day of the Extra Day: Since there is 1 day left after dividing the year into weeks, the day of the week that this extra day represents determines whether there will be 53 occurrences of that day in the year. For instance, if the year starts on a Sunday, then Sunday is the extra day.
3. Calculating the Probability: In a non-leap year of 365 days, every day of the week (Sunday, Monday, …, Saturday) has an equal probability of being the extra day. There are 7 possible choices for the extra day.
4. Probability of 53 Sundays: If the extra day is Sunday, then there will be 53 Sundays in that year. Therefore, the probability of having 53 Sundays is the fraction of favorable outcomes over the total number of outcomes, which is:

\(\frac{1}{7}\) because there is 1 favorable outcome (having Sunday as the extra day) out of 7 possible outcomes (any day of the week can be an extra day).

Thus, the correct answer is:
\(\frac{1}{7}\).