25. For an exam answer, a student can receive one of the following grades: $5,4,3,2$. The probability that a student will receive a grade of 5 is 0.3; a grade of $4-0.4$; a grade of $3-0.2$ and a grade of $2-0.1$. Which of the named events form a complete group of mutually exclusive events? What event is opposite to the event: “the student will receive a grade of 5” and what is the probability of this event?
S olution. Consider the events: $A_{1}, A_{2}, A_{3}, A_{4}$ - the student will receive, respectively, grades: $5,4,3,2$;

$A$ - the student will receive one of these grades: either 5, or 4, or 3, or 2.

The probabilities of events $A_{1}, A_{2}, A_{3}, A_{4}$ are: $P\left(A_{1}\right)=0.3 ; P\left(A_{2}\right)=$ $=0.4 ; P\left(A_{3}\right)=0.2 ; P\left(A_{4}\right)=0.1$.

$A_{1}, A_{2}, A_{3}, A_{4}$ - are mutually exclusive events forming a complete group. Event $A$ is the sum of these events:

$A=A_{1}+A_{2}+A_{3}+A_{4}$. Event $A$ is certain; $P(A)=1$.

The event $A_{1}$ - the student will receive a grade of 5, is opposite to the event $\bar{A}_{1}$ - the student will not receive a grade of 5. Using formula (11), we find: $P\left(\bar{A}_{1}\right)=1-P\left(A_{1}\right)=1-0.3=0.7$.