## 41. English Dictionary

In the class, there are 30 students. On each lesson, Miss Blackwick, the English teacher, assigns them to memorize a page of an English-French dictionary for the next class. At each lesson, she quizzes one student out of three, and among those questioned, there is always one out of ten who was already called on the previous lesson. One of the students, a notorious slacker, decided that he would only prepare the homework for the next lesson if he wasn't asked in the previous one.

"Which is more likely," he wonders, "to study in vain or to be punished by Miss Blackwick for not studying the lesson?"

Resolve his doubts.
41. Since one student out of three is called each time and among the called students, one out of ten was asked in the previous lesson, one student out of 30 is called for two consecutive days. Therefore, the risk of being punished by Miss Blackwick is one chance in 30. This means that a negligent student will be asked at least once in the next two consecutive lessons with a probability of

$$
1 / 3 + 1 / 3 - 1 / 30 = 19 / 30
$$

From this, it follows that he will not be asked in any of these lessons with a probability of

$$
1 - 19 / 30 = 11 / 30
$$

![](https://cdn.mathpix.com/cropped/2024_05_21_fe999c0fe2ad81fc5164g-123.jpg?height=634&width=691&top_left_y=1299&top_left_x=682)

This is the risk of studying the lesson in vain. It is 11 times greater than the risk of punishment ${ }^{1}$.