## 32. Rain and Good Weather

In a certain region, rainy days account for a quarter of all days. In addition, it has been observed that if it rains on a certain day, it will rain the next day in two out of three cases. What is the probability that the weather will be good on a given day if there was no rain the day before?
32. Let's consider the (n-1)th and nth days. There are four possible scenarios, the probabilities of which we will calculate.

## 116

The probability that it rains on both of these two days: \(1 / 4 \cdot 2 / 3 = 1 / 6\).

The probability that it rains on the nth day but not on the (n-1)th day: \(1 / 4 - 1 / 6 = 1 / 12\).

The probability that it rains on the (n-1)th day but not on the nth day: \(1 / 4 - 1 / 6 = 1 / 12\).

The probability that it does not rain on either of these two consecutive days: \(1 - 1 / 16 - 1 / 12 - 1 / 12 = 2 / 3\).

![](https://cdn.mathpix.com/cropped/2024_05_21_fe999c0fe2ad81fc5164g-118.jpg?height=548&width=557&top_left_y=711&top_left_x=744)

Therefore, the probability that the weather will be good on the nth day given that it did not rain on the (n-1)th day is

\[
2 / 3 : (1 - 1 / 4) = 8 / 9
\]