5. Anniversary of the Meeting (from 6th grade, 2 points). As is known, Robinson Crusoe's friend and companion was named Friday because they met on a Friday. This happened in January, but it is impossible to determine the exact year. Robinson is certain only that it could have happened in any year with equal probability, starting from 1668 and ending in 1671. Robinson and Friday revered the day of their meeting and celebrated it solemnly every year. What day of the week could the day of their eleventh anniversary have been? With what probability?
Solution. It is necessary to take into account that years can be common and leap years (if the year number is divisible by 4). A common year has 365 days, and a leap year has 366. When dividing by 7, the numbers 365 and 366 give remainders of 1 and 2, respectively. Therefore, if an event falls on a Friday in January of a non-leap year, it will be a Saturday exactly one year later. A leap year adds not one but two days of the week: if an event occurred on a Friday in January of a leap year, then exactly one year later it will be a Sunday.

Over 11 years, 11 days will accumulate, plus as many days as there were leap years during this period. If Friday and Robinson met in 1668, then over the next 11 years there were 3 leap years: 1668, 1672, and 1676. We need to add 14 days to Friday, which is exactly two weeks, so the 11th anniversary fell on a Friday again. Similarly, we will consider the other years. For convenience, let's create a table.

| Year of meeting | Number of leap years over 11 years | Days accumulated | Day of the week on the 11th anniversary |
| :--- | :--- | :---: | :--- |
| 1668 | $3(1668,1672$ and 1676$)$ | 14 | Friday |
| 1669 | $2(1672$ and 1676$)$ | 13 | Thursday |
| 1670 | $3(1672,1676$ and 1680$)$ | 14 | Friday |
| 1671 | $3(1672,1676$ and 1680$)$ | 14 | Friday |

Thus, the 11th anniversary of the meeting of Robinson and Friday fell on a Friday or a Thursday. The corresponding probabilities are 0.75 and 0.25.

Answer: on a Friday with a probability of 0.75 or on a Thursday with a probability of 0.25.

## $\mathrm{XI}$ Internet Olympiad on Probability Theory and Statistics. Main (correspondence) round. 15.12.77-21.01.18