Question

The probability that a leap year selected at random contains either 53 Sundays or 53 Mondays, is : (a) \(\frac{17}{53}\) (b) \(\frac{1}{53}\) (c) \(\frac{3}{7}\) (d) none of these

Short answer

Answer: The probability is \(\frac{3}{7}\).

Step-by-step solution

Determine the total number of possible cases

First, let's determine the total number of possible cases. Since the 1st of January in a leap year may fall on any of the seven days in a week, we have 7 possible cases in total.

Determine the number of favorable cases

Now, let's determine the favorable cases, i.e. the cases where the leap year will have 53 Sundays or 53 Mondays. Leap years have 366 days, and out of these, 52 full weeks account for 364 days, with the remaining 2 days being extras. 1. If the 1st of January is a Saturday, the extra 2 days will be a Sunday and a Monday. In this case, the leap year will have 53 Sundays and 53 Mondays. 2. If the 1st January is a Sunday, the extra 2 days will be a Sunday and a Monday. In this case, the leap year will have 53 Sundays and 53 Mondays. 3. If the 1st January is a Monday, the extra 2 days will be a Monday and a Tuesday. In this case, the leap year will have 53 Mondays. 4. In the rest of the cases (if the 1st of January is a Tuesday, Wednesday, Thursday or Friday), the leap year will not have 53 Sundays nor 53 Mondays. Hence, there are 3 favorable cases.

Calculate the probability

Finally, let's calculate the probability. The probability of a leap year having either 53 Sundays or 53 Mondays can be written as the ratio of the number of favorable cases to the total number of possible cases: \(P = \frac{\text{Number of favorable cases}}{\text{Total number of possible cases}} = \frac{3}{7}\) So, the correct answer is (c) \(\frac{3}{7}\).

Key concepts

Leap Year

A leap year has 366 days instead of the usual 365. This extra day is added to keep our calendar aligned with the Earth's revolutions around the Sun. Typically, it's added as February 29. Leap years come every four years, with some exceptions: a year is not a leap year if it's divisible by 100, unless it's also divisible by 400.

This means the years 2000 and 2400 are leap years, but 2100, 2200, and 2300 are not. Understanding the concept of a leap year helps in various real-world scenarios, including calculating days of the week over a year span. It introduces subtle changes like accommodating an extra day in those years which affects weekly scheduling.

Days of the Week

The days of the week are integral to understanding how they fit into a year. Normally, a week has 7 days, but over a year, this aligns as 52 weeks plus 2 or 3 extra days.

In a regular year, you have 52 weeks and one extra day. In a leap year, with 366 days, you encounter 52 weeks and 2 extra days. This additional day's placement varies depending on the starting weekday of the year.

• Affects events like which days will occur more frequently.
• Determines the likelihood of certain events, like having extra Sundays or Mondays.

The study of days in any specific year can be critical for planning and assessing frequency of events.

Favorable Cases

Favorable cases in probability are the specific scenarios that achieve the desired outcome. In the context of this problem, we are interested in the scenarios where a leap year has either 53 Sundays or 53 Mondays.

Leap years provide an interesting dynamic with their two extra days. For our problem, the key is:

• 1st January on a Saturday or Sunday leads to 53 Sundays and 53 Mondays.
• 1st January on a Monday results in 53 Mondays.

Thus, there are three favorable starting days for achieving our goal and these factor into calculating the probability of occurrence.

Ratio and Proportion

Ratio and proportion are mathematical tools used to compare two quantities and to understand their relationship. In probability, it's expressed as the ratio of favorable cases to possible cases.

To find the probability of our scenario—selecting a leap year with 53 Sundays or Mondays—we calculate:

• Total possible starting days: 7 (one for each day of the week).
• Favorable starting days: 3 (Saturday, Sunday, Monday).
• Probability: \( \frac{3}{7} \).

This ratio helps us conceptualize the likelihood or chance that a particular event occurs, critical for decision-making processes in various fields.