ANSWERS: 3
  • A 366-day leap year contains 52 weeks plus 2 days. To have an extra Sunday the leap year must begin on Friday or Saturday. One might assume that the days of the week are randomly distributed across the years, in which case the probability would be 2 days out of 7 = 2/7 = 0.285714... It turns out, however, that the days of the week follow a 400-year cycle (Ref: http://www.mcs.csuhayward.edu/~malek/Mathlinks/Weekdays.html). Since 400 is not a multiple of 7, there's a slightly uneven distribution of days of the week, with years beginning slightly more often on Friday than Saturday. Fortunately these deviations from the mean are opposite and cancel each other in this problem. So the final answer should be approximately 0.286.
  • Leap years have a pattern that repeats every 400 years. Days of the week have a pattern that repeats every 7 days. The pattern of the first day of a leap year certainly repeats every 7*400 years (if not more often). A leap year has 52 weeks of 7 days plus two more days. For a leap year to have two Sundays one of those two days must be a Sunday. The first(and 365th) or second(and 366th) must be a Sunday. If the second is a Sunday the first is a Saturday. In the 2800 year cycle, the first day of a leap year starts on the following days: Saturday: 91 times. Sunday: 105 times. Monday: 91 times Tuesday: 98 times Wednesday: 98 times Thursday: 91 times Friday: 105 times A total of 679 leap years. The leap year starts on Saturday or Sunday in 196 cases out of 679 leap years. So the probability is (196/679) or approximately 0.2887

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