Definitions and Properties of Probability

A content creator on TikTok decides to raffle three different prizes among their followers. To do this, they use a drum with balls numbered from $1$ to $20$. We consider the random experiment of extracting a ball and noting its number.

Define the following elements:

1. a) The sample space $E$.
2. b) The event $A$: "obtaining a number that is a multiple of $5$".
3. c) The event $B$: "obtaining a prime number greater than $10$".
4. d) The event $C$: "obtaining a number less than $4$".

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

We roll a six-sided die numbered from $1$ to $6$. We consider the events:

• $A = \{2, 4, 6\}$ (rolling an even number)

• $B = \{1, 2, 3\}$ (rolling less than or equal to $3$)

• $C = \{5\}$ (rolling a five)

Determine whether the following pairs of events are compatible or incompatible:

1. a) $A$ and $B$
2. b) $A$ and $C$
3. c) $B$ and $C$

Justify your answer by checking whether the intersection of each pair is the empty set $\emptyset$.

State with reasons 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) $P(x_1) = 1; \quad P(x_2) = 0; \quad P(x_3) = 0$

In a high school, $25\%$ of the students are enrolled in guitar lessons, and $15\%$ are enrolled in piano lessons. If we know that there are no students studying both instruments at the same time:

1. a) What are these types of events called in probability?
2. b) Calculate the probability that a student chosen at random studies guitar or piano.
3. c) Calculate the probability that a student does not study either of these two instruments.

In an urn there are $12$ colored balls: $5$ are red, $4$ are blue and $3$ are yellow. We extract a ball at random and observe its color. 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 card is drawn from a Spanish deck of $40$ cards. Calculate the probability of obtaining:

1. a) A king.
2. b) A card of cups.
3. c) The ace of coins.
4. d) A face card (jack, knight or king).
5. e) A card that is not an ace.

In a prestigious esports tournament, a digital doping control committee made up of 4 members must be chosen. The group of candidates consists of 8 data analysts and 6 cybersecurity experts. If the selection is made at random, calculate the probability of the following events:

1. a) The committee is made up exclusively of cybersecurity experts.
2. b) There is at least one representative of each profile on the committee.
3. c) The number of data analysts is strictly greater than that of cybersecurity experts.

A music app has analyzed the preferences of a group of young people. It is known that the probability of a user liking Trap is $P(T) = 0,65$, the probability of them liking Pop is $P(P) = 0,40$ and the probability of them liking 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 science fiction 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.

In a charity raffle organized by students for the end-of-year trip, tickets numbered from $000$ to $999$ are sold. We consider the events:

$A =$ "The number ends in an even digit (0, 2, 4, 6, 8)".

$B =$ "The number is a multiple of 25".

a) Calculate $P(A)$ and $P(B)$ by applying Laplace's rule.

b) Are $A$ and $B$ compatible or incompatible events? Justify your answer by finding the event $A \cap B$.

c) Calculate the probability that the winning number ends in an even digit or is a multiple of 25.

Observe the following Venn diagram representing the events $A$ and $B$ within a sample space $E$. The numbers represent the quantity of elementary events in each region.

If the experiment consists of choosing at random one of the elementary events of $E$, all of them equally likely, calculate:

1. a) $P(A)$
2. b) $P(A \cap B)$
3. c) $P(A \cup B)$
4. d) $P(\bar{A} \cap \bar{B})$ (that it is neither in $A$ nor in $B$).

A market study on the consumption of digital platforms among 16-year-olds reveals that $70\%$ use Spotify, $55\%$ use TikTok and $45\%$ use Netflix. Furthermore, it is known that $30\%$ use Spotify and TikTok, $25\%$ use Spotify and Netflix, and $20\%$ use TikTok and Netflix. Only $10\%$ of the young people use all three platforms simultaneously.

If we select a young person at random, calculate and explain the probability that they:

1. a) Use any of the three platforms.
2. b) Use none of these platforms.
3. c) Use only one of the three platforms.

A survey has been carried out among students at a high school about the 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 of the two 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? Justify your answer.

Given two events $A$ and $B$ associated with the same random experiment, we know that the probability that neither of them 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 with reasoning, 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) $P(\bar{A} \cup \bar{B})$

In a language academy, students are divided according to the level they are studying and the language chosen. The data is collected in the following contingency table:

a) Complete the missing data in the table.

b) If we choose a student at random, calculate the probability that they study French or are in level B2.

c) Given that the chosen student is in level B1, what is the probability that they study English?

A box contains 10 wireless earphones, of which 3 are defective. If we extract two earphones consecutively and without replacement, calculate the probability that:

a) Both earphones work correctly.

b) At least one of the earphones is defective.

c) The second earphone is defective given that the first one was not.

An esports team participates in a tournament with two independent knockout stages. The probability of passing the first stage is $0,7$ and the probability of passing the second is $0,6$. Represent the situation using a tree diagram and calculate:

a) The probability that the team wins both stages.

b) The probability that they are eliminated in either of the two stages.

c) The probability that they pass exactly one of the two stages.

Let $A$ and $B$ be two events associated with a random experiment whose probabilities are known: $P(A) = 0,4$, $P(B) = 0,5$ and $P(A \cup B) = 0,7$.

a) Calculate $P(A \cap B)$.

b) Determine whether $A$ and $B$ are independent events.

c) Calculate the probability of the union of the complementary events, that is, $P(\bar{A} \cup \bar{B})$.

Determine, with reasoning, whether the following statements are true or false, providing a numerical counterexample if they are false:

1. a) If two events $A$ and $B$ are mutually exclusive ($A \cap B = \emptyset$), then they must necessarily be dependent.
2. b) If $P(A \cup B) = P(A) + P(B)$, then the events $A$ and $B$ are independent.
3. c) If $P(A) = 0,6$ and $P(B) = 0,5$, then the events $A$ and $B$ cannot be mutually exclusive.

In a high school, $65\%$ of the students play soccer ($F$) and $50\%$ play basketball ($B$). Knowing that the probability that a student plays neither of the two sports is $P(\bar{F} \cap \bar{B}) = 0,15$:

1. a) Determine whether the events $F$ and $B$ are independent.
2. b) If the chess club ($A$) is an event such that $A \subset F$ and $P(A) = 0,2$, calculate the probability that a student chosen at random plays soccer but not chess.
3. c) Calculate $P(\bar{F} \cup \bar{B})$ and interpret its meaning in the context of the problem.

In a youth health center, $30\ \%$ of the patients attend for nutritional reasons ($N$), $50\ \%$ for mental health ($S$) and $20\ \%$ for routine check-ups ($R$). It is known that $10\ \%$ of those attending for nutrition, $20\ \%$ for mental health and $5\ \%$ for check-ups are over 18 years old ($M$).

a) Calculate the probability that a patient chosen at random is over 18 years old.

b) If we choose a patient and they turn out to be over 18 years old, what is the probability that they attended for mental health reasons?

A computer security system has three independent protection layers ($C_1, C_2, C_3$). The probability that an attacker manages to breach the first layer is $0,15$; the probability of breaching the second is $0,10$, and that of the third is $0,05$. The system is only considered compromised if the attacker manages to breach all three layers.

1. a) What is the probability that the system is compromised?
2. b) What is the probability that the attacker breaches at least one protection layer?
3. c) What is the probability that the attacker breaches exactly two of the three layers?

A six-sided die is rolled which has been biased in such a way that the even faces all have the same probability of appearing, and the odd faces also have the same probability among themselves. However, it is known that the probability of obtaining an even number is three times that of obtaining an odd number.

1. a) Determine the probability of each of the six faces of the die.
2. b) If the die is rolled twice, what is the probability that the sum of the scores obtained is a prime number?
3. c) Calculate the probability of obtaining at least one number greater than 4 in three consecutive rolls.

In a biological laboratory, DNA samples are being analyzed. It has been observed that in an experiment with 5 independent samples, the probability that at least one sample presents a specific mutation is $0,67232$.

1. a) Calculate the probability that an individual sample chosen at random presents this mutation (assuming it is the same for all samples).
2. b) If 3 samples are analyzed, what is the probability that exactly two present the mutation?
3. c) How many samples would have to be analyzed, as a minimum, for the probability of finding at least one mutation to be greater than $0,95$?