Question:

Step 1: Determine the number of years in the 22nd century and classify them as leap or non-leap years. The 22nd century spans from the year 2101 to 2200, inclusive. This period contains 100 years. A year is a leap year if it is divisible by 4, unless it is a century year not divisible by 400. In the 22nd century, the years divisible by 4 are 2104, 2108, ..., 2196, and 2200. The year 2200 is a century year. Since 2200 is not divisible by 400, it is not a leap year. Therefore, the leap years in the 22nd century are 2104, 2108, ..., 2196. The number of leap years is $\frac{2196 - 2104}{4} + 1 = \frac{92}{4} + 1 = 23 + 1 = 24$. The number of non-leap years is $100 - 24 = 76$. Step 2: Determine the probability of a non-leap year having 53 Sundays. A non-leap year has 365 days. Since $365 = 52 \times 7 + 1$, a non-leap year consists of 52 full weeks and 1 additional day. For a non-leap year to have 53 Sundays, this additional day must be a Sunday. This occurs if the year starts on a Sunday. Assuming the starting day of a randomly chosen year is uniformly distributed among the 7 days of the week, the probability of a non-leap year having 53 Sundays is $\frac{1}{7}$. Step 3: Determine the probability of a leap year having 53 Sundays. A leap year has 366 days. Since $366 = 52 \times 7 + 2$, a leap year consists of 52 full weeks and 2 additional days. For a leap year to have 53 Sundays, at least one of these two additional days must be a Sunday. This occurs if the year starts on a Saturday or a Sunday. Assuming the starting day of a randomly chosen year is uniformly distributed among the 7 days of the week, the probability of a leap year having 53 Sundays is $\frac{2}{7}$. Step 4: Calculate the total probability of a year in the 22nd century having 53 Sundays. The probability is the weighted average of the probabilities for non-leap and leap years: $$P(\text{53 Sundays}) = P(\text{non-leap year}) \times P(\text{53 Sundays | non-leap}) + P(\text{leap year}) \times P(\text{53 Sundays | leap})$$ $$P(\text{53 Sundays}) = \left(\frac{76}{100} \times \frac{1}{7}\right) + \left(\frac{24}{100} \times \frac{2}{7}\right)$$ $$P(\text{53 Sundays}) = \frac{76}{700} + \frac{48}{700}$$ $$P(\text{53 Sundays}) = \frac{76 + 48}{700}$$ $$P(\text{53 Sundays}) = \frac{124}{700}$$ To simplify the fraction, divide the numerator and denominator by their greatest common divisor, which is 4: $$P(\text{53 Sundays}) = \frac{124 \div 4}{700 \div 4} = \frac{31}{175}$$ The probability that in a year of the 22nd century, chosen at random, there will be 53 Sundays, is $\frac{31}{175}$.