Question

The probability that a leap year will have 53 Sunday or 53 Monday is (a) \((2 / 7)\) (b) \((3 / 7)\) (c) \((4 / 7)\) (d) \((1 / 7)\)

Short answer

The probability of a leap year having 53 Sundays or 53 Mondays is (b) \(\frac{3}{7}\).

Step-by-step solution

Determine the number of occurrences of each day in a leap year

In a leap year, there are 366 days, consisting of 52 weeks plus 2 extra days. Since each day of the week occurs 52 times, we need to focus on determining the probability of the extra 2 days being Sunday and Monday.

Calculate the probability of having 53 Sundays in a leap year

To have 53 Sundays in a leap year, one of the extra two days must fall on a Sunday. There are 7 possibilities for the two extra days (Sunday & Monday, Monday & Tuesday, Tuesday & Wednesday, etc.). To have 53 Sundays, we need the extra days to be one of the following: (Sunday & Monday), (Saturday & Sunday). Therefore, there are 2 favorable cases out of 7 total possibilities: \[P(53\,Sundays) = \frac{2}{7}\]

Calculate the probability of having 53 Mondays in a leap year

Similarly, to have 53 Mondays in a leap year, one of the extra days must fall on Monday. There are also 7 possibilities for the extra 2 days. To have 53 Mondays, we need the extra days to be one of the following: (Sunday & Monday), (Monday & Tuesday). There are also 2 favorable cases out of 7 total possibilities: \[P(53\,Mondays) = \frac{2}{7}\]

Combine the probabilities

However, we can't just add these probabilities together because there is an overlapping condition (having the extra days as Sunday & Monday) which leads to having both 53 Sundays and 53 Mondays. To avoid double-counting this overlapping condition, we need to consider three cases: Case 1: There are 53 Sundays and not 53 Mondays, i.e. (Saturday & Sunday) Case 2: There are 53 Mondays and not 53 Sundays, i.e. (Monday & Tuesday) Case 3: There are 53 Sundays and 53 Mondays, i.e. (Sunday & Monday) There are 3 favorable cases out of 7 possibilities, so the probability of a leap year having 53 Sundays or 53 Mondays is: \[P(53\,Sundays\,or\,53\,Mondays) = \frac{3}{7}\] Therefore, the correct answer is (b) \((3 / 7)\).

Key concepts

Probability Calculation

Understanding probability calculation is essential when determining the likelihood of different outcomes. Simply put, probability measures the chance that a given event will occur, and it is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.

For example, in the context of a leap year, which comprises 366 days instead of the usual 365, we want to calculate the probability of having either 53 Sundays or Mondays. The probability of an event E, denoted as P(E), is calculated using the formula:
\[\begin{equation}P(E) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}}\end{equation}\]
In the leap year scenario, we would use this formula to calculate the chance of having 53 Sundays or Mondays by identifying the favorable outcomes (e.g., combinations of the extra two days that result in an extra Sunday or Monday) and dividing by the total number of possible outcomes for these two days.

Extra Days in Leap Year

A leap year is a year containing one additional day in order to keep the calendar year synchronized with the astronomical or seasonal year. This is crucial because the Earth's orbit around the Sun takes approximately 365.25 days, so an extra day—February 29th—is added every four years to account for this quarter day. As a result, a leap year has 366 days.

For the purpose of probability calculations with leap years, the focus is typically on the 'extra two days.' This is because while most years have 52 Sundays and 52 Mondays, a leap year's extra two days can result in 53 Sundays or Mondays. Understanding how these extra days fall within the leap year is crucial for determining the probability of having 53 Sundays or Mondays.

Combinatorial Probability

Combinatorial probability involves determining the likelihood of an event by considering all possible combinations that could lead to that event. It is often used when we need to know the probability of various outcomes where order and combination matter.

When considering a leap year, the combinatorial aspect comes into play with the extra two days. There are seven different combinations in which those two days can occur when considering the sequence of days in a week. For example, the extra days could be (Sunday & Monday), (Monday & Tuesday), (Tuesday & Wednesday), and so on, up to (Saturday & Sunday).

In calculating the probability of having 53 Sundays or 53 Mondays, we count the combinations that include an extra Sunday or an extra Monday. We must be mindful of overlapping cases; for instance, the combination (Sunday & Monday) counts as an extra day for both Sunday and Monday. Combinatorial probability allows for a structured way to account for these scenarios to provide an accurate probability measure.