MHTCET 15 May 2024 Shift 1 Paper

Concept:The probability depends on whether the selected year is a leap year or a non-leap year, and on the distribution of the extra days.Explanation:A randomly selected year is a leap year with probability $\frac{1}{4}$.So the probability of a non-leap year is $1 - \frac{1}{4} = \frac{3}{4}$.A non-leap year has 52 weeks and 1 extra day.That extra day can be any of the 7 days of the week, so the chance it is Monday is $\frac{1}{7}$.A leap year has 52 weeks and 2 extra days.The 2 extra days can be the pairs: (Mon, Tue), (Tue, Wed), (Wed, Thu), (Thu, Fri), (Fri, Sat), (Sat, Sun), (Sun, Mon).Out of these 7 pairs, 2 pairs contain Monday: (Mon, Tue) and (Sun, Mon).So the chance of 53 Mondays in a leap year is $\frac{2}{7}$.Therefore, the required probability is:$$\frac{3}{4} \times \frac{1}{7} + \frac{1}{4} \times \frac{2}{7} = \frac{3}{28} + \frac{2}{28} = \frac{5}{28}$$Answer:The probability is $\frac{5}{28}$, which corresponds to option C.