In a leap year, the probability of getting 53 Sundays is

Concept: A common year (365 days = 52 weeks + 1 day) guarantees each weekday 52 occurrences, with a 53rd only if the single extra day matches that weekday. A leap year (366 days = 52 weeks + 2 days) instead leaves two extra days, which are always the two consecutive weekdays right after the 52 complete weeks -- one of 7 equally likely pairs depending on which day the year begins.

Application:

1. 366 divided by 7 gives 52 remainder 2, so 366 days = 52 complete weeks + 2 extra days.

2. Since the year's calendar runs as one continuous 366-day cycle, the two extra days are always the two weekdays right after the 52 complete weeks -- this pair is fixed by the weekday the year starts on.

3. As the year can start on any of the 7 weekdays with equal likelihood, the extra-day pair takes one of 7 equally likely forms: (Monday, Tuesday), (Tuesday, Wednesday), (Wednesday, Thursday), (Thursday, Friday), (Friday, Saturday), (Saturday, Sunday), (Sunday, Monday).

4. A 53rd Sunday occurs precisely when Sunday is one of these two extra days -- this happens for exactly 2 of the 7 pairs: (Saturday, Sunday) when the year starts on Saturday, and (Sunday, Monday) when the year starts on Sunday.

5. So the probability of getting 53 Sundays in a leap year is 2/7.

Cross-check: Testing every starting weekday directly: starting Monday through Friday all give extra-day pairs with no Sunday, while starting on Saturday gives (Saturday, Sunday) and starting on Sunday gives (Sunday, Monday), both containing a Sunday. Exactly 2 of the 7 starting days work, confirming a probability of 2/7.