31. In the case, there are 20 cassettes with disco music recordings and 10 - with techno music recordings. The DJ randomly takes two cassettes one after another. What is the probability that
1) the first cassette has disco music recorded on it;
2) the second cassette also has disco music recorded on it. Consider two cases: before taking the second cassette, the DJ

a) returns the first cassette to the case;

b) does not return the first cassette to the case.
Solution. Consider the events:

$A$ - disco music is recorded on the first cassette;

$B$ - disco music is recorded on the second cassette.

1) Let's find the probability of event $A$, using the classical definition of probability. The total number of outcomes of the experiment $n=30$; the number of outcomes of the experiment favorable to event $A$ is $m=20$; the required probability $P(A)=m / n=20 / 30=2 / 3$.

2, a) Event $B$ is independent of event $A$, since the occurrence or non-occurrence of event $A$ does not change the probability of event $B$ occurring. The number of outcomes of the experiment favorable to event $B$ is $m_{1}=20$; the total number of outcomes of the experiment $n=30$; therefore, $P(B)=m_{1} / n=20 / 30=2 / 3$.

2, b) Event $B$ is dependent on event $A$, since the occurrence or non-occurrence of event $A$ changes the probability of event $B$ occurring. We need to find the probability of event $B$ given that the cassette taken from the case first also has disco music recorded on it, that is, we need to find $P_{A}(B)$ - the conditional probability of event $B$ calculated given that event $A$ has occurred. Since the first cassette taken is not returned to the case, the number of outcomes of the experiment favorable to event $B$ is $m_{2}=19$, and the total number of outcomes of the experiment is $n_{1}=29$. The required conditional probability $P_{A}(B)=m_{2} / n_{1}=19 / 29$.