Question

A probability that in a year of the $22$nd century chosen at random, there will be $53$ Sundays is

• A. $\frac{3}{28}$
• B. $\frac{2}{28}$
• C. $\frac{7}{28}$
• D. $\frac{5}{28}$

Answer 1

The correct option is D

$\frac{5}{28}$

Explanation for the correct option:

Step 1: Find the probability of a leap year:

Estimate the probability that the selected year is a leap year in the century.

As in every four years, there is a leap year.

So in $100$ years, there will be $25$ leap years.

$\begin{array}{rcl}p& =& \frac{25}{100}\\ & =& \frac{1}{4}\end{array}$

Step 2: Find the probability that out of the $2$ extra days of the leap year, one is Sunday:

Out of $7$days, chances of having Sunday occupying one of those extra two days are $2$.

So, $p=\frac{2}{7}$

So, the probability of having Sunday on a leap year $=\frac{1}{4}\times \frac{2}{7}$

Step 3: Find the probability of having $53$ Sundays on a normal year:

A normal year has an extra day.

The probability that Out of $7$days, chances of having Sunday occupying that day $=\frac{1}{7}$

So, the probability of having Sunday on a normal year $=\frac{3}{4}\times \frac{1}{7}$

So, the total probability$$\begin{gathered} =\frac{1}{4}\times \frac{2}{7}+\frac{3}{4}\times \frac{1}{7} \\ =\frac{5}{28} \end{gathered}$$

Hence, option (D) is the correct answer.