MathDB

3

Part of 1975 Poland - Second Round

Problems(1)

probability to wash the dishes

Source: Polish MO Second Round 1975 p3

9/8/2024
In a certain family, a husband and wife made the following agreement: If the wife washes the dishes one day, the husband washes the dishes the next day. However, if the husband washes the dishes one day, then who washes the dishes the next day is decided by drawing a coin. Let pn p_n denote the probability of the event that the husband washes the dishes on the n n -th day of the contract. Prove that there is a limit lim⁡n→∞pn \lim_{n\to \infty} p_n and calculate it. We assume p1=12 p_1 = \frac{1}{2} .
combinatoricsprobability