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

The correct option is 5/7.

Let us break down the logical reasoning step-by-step to understand why this is the correct probability:

1. Understanding a leap year:
A normal year has 365 days, whereas a leap year has 366 days.

2. Dividing the days into weeks:
A week consists of 7 days. Let us divide the total number of days in a leap year by 7:

$$\frac{366}{7}=52\text{weeks and}2\text{extra days}$$

This calculation shows that a leap year contains 52 complete weeks and exactly 2 extra days.
Since there are 52 complete weeks, there will definitely be 52 Tuesdays (as well as 52 of every other day of the week) in a leap year.

3. Analyzing the 2 extra days:
To get a 53rd Tuesday, one of the 2 extra days must be a Tuesday. Let us list all the possible consecutive pairs of days that these 2 extra days can be:

• (Sunday, Monday)
• (Monday, Tuesday)
• (Tuesday, Wednesday)
• (Wednesday, Thursday)
• (Thursday, Friday)
• (Friday, Saturday)
• (Saturday, Sunday)

There are a total of 7 possible outcomes. Let this be the total number of outcomes:
$n\left(S\right)=7$

4. Finding the probability of GETTING 53 Tuesdays:
Out of the 7 pairs listed above, the pairs that contain a Tuesday are:
1. (Monday, Tuesday)
2. (Tuesday, Wednesday)

So, there are 2 favorable outcomes for getting 53 Tuesdays:
$n\left(E\right)=2$

Therefore, the probability of getting 53 Tuesdays in a leap year, denoted by P(E), is:

$$P\left(E\right)=\frac{n\left(E\right)}{n\left(S\right)}=\frac{2}{7}$$

5. Finding the probability of NOT getting 53 Tuesdays:
The event of "not getting 53 Tuesdays" is the complement of event E, denoted as E'.
The probability of a complementary event is calculated as:

$$P\left(E^{\prime }\right)=1-P\left(E\right)$$

Substituting the value of P(E):

$$P\left(E^{\prime }\right)=1-\frac{2}{7}=\frac{7-2}{7}=\frac{5}{7}$$

Thus, the probability of not getting 53 Tuesdays in a leap year is 5/7.