Definitions

A high school drama club decides to raffle three tickets for its premiere among the audience. To do so, it uses a drum containing balls numbered from $1$ to $20$. Consider the random experiment of drawing one ball and noting its number.

Define the following elements:

1. a) The sample space $E$.

1. b) The event $A$: "getting a number that is a multiple of $5$".

1. c) The event $B$: "getting a prime number greater than $10$".

1. d) The event $C$: "getting a number less than $4$".

Write all the elementary events that make up each of the sets above.

In an urn there are $12$ coloured balls: $5$ are red, $4$ are blue and $3$ are yellow. We draw a ball at random and observe its colour. Using Laplace's rule, calculate the probability of the following events:

1. a) $P(R)$: "the ball is red".
2. b) $P(A \cup M)$: "the ball is blue or yellow".
3. c) $P(\bar{R})$: "the ball is not red".
4. d) $P(V)$: "the ball is green".

A music app has analysed the preferences of a group of young people. It is known that the probability that a user likes Trap is $P(T) = 0,65$, the probability that they like Pop is $P(P) = 0,40$ and the probability that they like both genres is $P(T \cap P) = 0,15$.

Calculate the probability that, when choosing a user at random:

1. a) They like Trap or Pop.
2. b) They do not like Trap.
3. c) They like Trap but not Pop ($P(T - P)$).

On a streaming platform, $65\ \%$ of the subscribers have watched the latest season of a sci-fi series ($S$), $40\ \%$ have watched a nature documentary ($D$) and $25\ \%$ have watched both. If we select a user at random, calculate the probability of the following events:

a) That they have watched the series or the documentary.

b) That they have watched neither the series nor the documentary.

c) That they have watched the series but not the documentary.

State with justification whether the following probability assignments for a sample space $E = \{x_1, x_2, x_3\}$ are valid according to Kolmogorov's axioms:

1. a) $P(x_1) = 0,4; \quad P(x_2) = 0,3; \quad P(x_3) = 0,3$
2. b) $P(x_1) = 0,7; \quad P(x_2) = -0,1; \quad P(x_3) = 0,4$
3. c) $P(x_1) = 0,5; \quad P(x_2) = 0,5; \quad P(x_3) = 0,2$
4. d) $P(x_1) = 1; \quad P(x_2) = 0; \quad P(x_3) = 0$

A survey was conducted among secondary school students regarding their use of social media. It is known that the probability that a student uses Instagram ($I$) is $P(I) = 0,82$ and the probability that they use TikTok ($T$) is $P(T) = 0,55$. If the probability that they use neither is $P(\bar{I} \cap \bar{T}) = 0,10$, use De Morgan's laws to calculate:

a) The probability that they use at least one of the two social media platforms.

b) The probability that they use both simultaneously.

c) Are the events "using Instagram" and "using TikTok" incompatible? Give reasons for your answer.

Given two events $A$ and $B$ associated with the same random experiment, we know that the probability that neither occurs is $P(\bar{A} \cap \bar{B}) = 0,2$. Furthermore, it is known that $P(A) = 0,5$ and that the probability that $A$ occurs but $B$ does not occur is $P(A - B) = 0,3$.

Calculate step-by-step, using the properties of probability and De Morgan's laws:

1. a) $P(A \cup B)$
2. b) $P(A \cap B)$
3. c) $P(B)$
4. d) $P(\bar{A} \cup \bar{B})$