Question

The probability of not getting 53 Tuesdays in a leap year is:

• A. \(\frac{2}{7}\)
• B. \(\frac{1}{7}\)
• C. 0
• D. \(\frac{5}{7}\)

Hint:

When calculating probabilities in scenarios involving days of the week or cycles, it's useful to identify the total number of possible outcomes and then determine the favorable outcomes. In this case, the extra days in a leap year determine whether there will be an additional Tuesday. By listing the possible pairs of extra days, you can easily count the favorable cases and compute the probability accordingly.

Answer 1

The Correct Option is D

In a leap year, there are 366 days, which consist of 52 complete weeks plus 2 extra days. 
These 2 extra days can be:

• Sunday and Monday
• Monday and Tuesday
• Tuesday and Wednesday
• Wednesday and Thursday
• Thursday and Friday
• Friday and Saturday
• Saturday and Sunday

To have 53 Tuesdays in a leap year, one of the two extra days must be a Tuesday. Thus, favorable outcomes are:

• Monday and Tuesday
• Tuesday and Wednesday

So, the probability of getting 53 Tuesdays is: \(\frac{2}{7}\)
Thus, the probability of not getting 53 Tuesdays is: \(\frac{5}{7} = 1 - \frac{2}{7}\)
This confirms the probability of not getting 53 Tuesdays in a leap year is: \(\frac{5}{7}\)

Answer 2

In a leap year, there are 366 days. This is equivalent to:

366 days = 52 full weeks + 2 extra days.
These extra days can be any pair of consecutive days: Sunday-Monday, Monday-Tuesday, Tuesday-Wednesday, Wednesday-Thursday, Thursday-Friday, Friday-Saturday, or Saturday-Sunday.

Counting the number of Tuesdays:

Since each full week contains exactly one Tuesday, there are 52 Tuesdays in a leap year.
The extra two days can either be Sunday-Monday, Monday-Tuesday, Tuesday-Wednesday, Wednesday-Thursday, Thursday-Friday, Friday-Saturday, or Saturday-Sunday.
In two of these pairs—Monday-Tuesday and Tuesday-Wednesday—there is an additional Tuesday. Therefore, there are two possible cases where there can be 53 Tuesdays in a leap year.

Probability of not getting 53 Tuesdays:

To avoid having 53 Tuesdays, the extra two days must not include a Tuesday. The possible pairs of extra days where this condition holds are: Sunday-Monday, Wednesday-Thursday, Thursday-Friday, Friday-Saturday, and Saturday-Sunday. There are 5 such pairs out of the 7 total possibilities.

Thus, the probability of not getting 53 Tuesdays is:

\[ P(\text{not getting 53 Tuesdays}) = \frac{5}{7}. \]

The correct answer is: \[ \frac{5}{7}. \]