## 29. Umbrella

One Saturday evening, two brothers, Jean and Pierre, exchanged their observations about the behavior of their neighbor, Madame Martin. They agreed that every Sunday she goes out of the house once (and only once), that in two out of three cases she takes an umbrella with her, and therefore even on good weather days, she goes for a walk with an umbrella half the time. However, Jean believed that on rainy days she never forgets to take an umbrella, while Pierre, on the contrary, claimed that sometimes she goes for a walk without an umbrella even on rainy days.

Which of the brothers, in your opinion, is right and which is wrong, if in their region, on average, half of the days are rainy?
29. Since Madame Martin takes an umbrella with her one time out of two when the weather is good, and good weather occurs once every two days, the event "she takes an umbrella with her when the weather is good" happens once every four days $(1 / 2 \cdot 1 / 2=1 / 4)$. However, it is known that Madame Martin generally takes an umbrella with her two times out of three. Therefore, the event "she takes an umbrella with her when the weather is bad" will occur five times out of twelve ( $2 / 3-1 / 4=5 / 12$ ). But since bad weather occurs once every two days, one time out of twelve ( $1 / 2-5 / 12=1 / 12$ ) Madame Martin leaves her umbrella at home when the weather is bad, i.e., Pierre was right, not Jean.