172. After the student answers the questions in the examination ticket, the examiner asks the student additional questions. The teacher stops asking additional questions as soon as the student shows ignorance of the given question. The probability that the student will answer any additional question is 0.9. Required: a) construct the distribution law of the random discrete variable $X$ - the number of additional questions the teacher will ask the student; b) find the most probable number $k_{0}$ of additional questions asked to the student.
Solution. a) The discrete random variable $X$ - the number of additional questions asked - has the following possible values: $x_{1}=1, x_{2}=2, x_{3}=3, \ldots, x_{k}=k, \ldots$ Let's find the probabilities of these possible values.

The variable $X$ will take the possible value $x_{i}=1$ (the examiner will ask only one question) if the student does not answer the first question. The probability of this possible value is $1-0.9=0.1$. Thus, $P(X=1)=0.1$.

The variable $X$ will take the possible value $x_{2}=2$ (the examiner will ask only two questions) if the student answers the first question (the probability of this event is 0.9) and does not answer the second (the probability of this event is 0.1). Thus, $P(X=2)=0.9 \cdot 0.1=0.09$.

Similarly, we find

$$
P(X=3) \leftrightharpoons 0.9^{2} \cdot 0.1=0.081, \ldots, P(X=k)=0.9^{k-1} \cdot 0.1, \ldots
$$

Let's write the required distribution law:

| $X$ | 1 | 2 | 3 | $\ldots$ | $k$ | $\ldots$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $p$ | 0.1 | 0.09 | 0.081 | $\ldots$ | $0.9^{k-1} \cdot 0.1$ | $\ldots$ |

b) The most probable number $k_{0}$ of questions asked (the most probable possible value of $X$), i.e., the number of questions asked by the teacher, which has the highest probability, as follows from the distribution law, is one.