Question

The probability, that a year selected at random will have 53 Mondays, is

• A. 1/7
• B. 2/7
• C. 3/28
• D. 5/28

Answer 1

Answer: (D)

5/28

Answer 2

We need to consider two cases since a “year selected at random” may be either a non‑leap year or a leap year.

Step 1. Non‑leap year (365 days):

A non‑leap year has 365 days = 52 weeks + 1 extra day. Only the day corresponding to the extra day will occur 53 times. Hence, the probability that the extra day is Monday is

Probability = 1/7.

Step 2. Leap year (366 days):

A leap year has 366 days = 52 weeks + 2 extra days. The two extra days are consecutive. For Monday to appear 53 times, Monday must fall as one of these extra days. That happens if the leap year starts on either Sunday or Monday. Thus, the probability = 2/7.

Step 3. Combining the cases:

Assuming the probability that a randomly chosen year is a leap year is 1/4 (and non‑leap is 3/4), we have

Overall probability = (Probability of non‑leap year × 1/7) + (Probability of leap year × 2/7) = (3/4 × 1/7) + (1/4 × 2/7) = (3/28) + (2/28) = 5/28.

Thus, the probability that a randomly selected year has 53 Mondays is 5/28.