32. Using the original data from the previous problem, find the probability that music in the disco style is recorded on two randomly taken cassettes in succession. Consider two cases: the DJ, before taking the second cassette,

a) returns the first cassette to the case;

b) does not return the first cassette to the case.
S o l u t i o n. Let's consider the events:

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

$\boldsymbol{B}$ - disco music is recorded on the second cassette;

C - disco music is recorded on two consecutively taken cassettes.

Event $C$ consists of disco music being recorded on both the first and second cassettes taken from the case. This means that event $C$ is the product of events $A$ and $B: C=A B$.

Let's find the probability of event $C$ by applying the multiplication theorem of probabilities.
a) $A$ and $B$ are independent events. Using formula (13), we get

$$
P(C)=P(A B)=P(A) \cdot P(B)=\frac{2}{3} \cdot \frac{2}{3}=\frac{4}{9} \approx 0.4444
$$

b) Event $B$ is dependent on event $A$. Using formula (12), we find

$$
P(C)=P(A B)=P(A) \cdot P_{A}(B)=\frac{20}{30} \cdot \frac{19}{29} \approx 0.4368
$$