Definition and properties of probability

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.

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

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

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

• $C = \{5\}$ (getting 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 $\emptyset$.

In an urn there are $12$ colored balls: $5$ are red, $4$ are blue and $3$ are yellow. We draw 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 getting:

1. a) A king.
2. b) A cups card.
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 anti-doping control committee consisting of 4 members must be chosen. The group of candidates is made up 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 consists 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 the number of cybersecurity experts.

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 ($P(T - P)$).

In a high school, $25\%$ of 12th grade students are enrolled in guitar lessons, and $15\%$ are enrolled in piano lessons. Given that no student studies both instruments at the same time:

1. a) What are these types of events called in probability?
2. b) Calculate the probability that a randomly chosen student studies guitar or piano.
3. c) Calculate the probability that a student studies neither of these two instruments.

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.

A study on the hobbies of 16-year-olds reveals that $60\%$ practice a sport, $55\%$ play a musical instrument, and $45\%$ go to the cinema regularly. Furthermore, it is known that $30\%$ play sports and play an instrument, $25\%$ play sports and go to the cinema, and $20\%$ play an instrument and go to the cinema. Only $10\%$ of the young people do all three activities simultaneously.

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

1. a) Engage in at least one of the three activities.
2. b) Do not engage in any of these activities.
3. c) Engage in only one of the three activities.

Observe the following Venn diagram representing events $A$ and $B$ within a sample space $E$. The numbers represent the number 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 is neither in $A$ nor in $B$).

A survey was conducted among high 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" mutually exclusive? Justify 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) $P(\bar{A} \cup \bar{B})$

In a language academy, students are divided according to the level they are studying and the language chosen. The data are 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 at level B2.

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

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, $P(\bar{A} \cup \bar{B})$.

Show with justification 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 incompatible ($A \cap B = \emptyset$), then they must necessarily be dependent.
2. b) If $P(A \cup B) = P(A) + P(B)$, then events $A$ and $B$ are independent.
3. c) If $P(A) = 0.6$ and $P(B) = 0.5$, then events $A$ and $B$ cannot be incompatible.

A box contains 10 wireless earphones, 3 of which are defective. If we take out 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 knockout rounds. The probability of getting through the first round is $0.7$ and, if they advance, the probability of getting through the second is $0.6$. Represent the situation using a tree diagram and calculate:

a) The probability that the team wins both rounds.

b) The probability that it is eliminated in either of the two rounds.

c) The probability that it gets through exactly one of the two rounds.

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 $P(B)$ using 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.

An IT security system has three independent layers of protection ($C_1, C_2, C_3$). 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 layer of protection?
3. c) What is the probability that the attacker breaches exactly two of the three layers?

In a youth health center, $30\ \%$ of 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 randomly chosen patient 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?

In a school, $65\%$ of the students play soccer ($F$) and $50\%$ play basketball ($B$). Given that the probability that a student plays neither of these two sports is $P(\bar{F} \cap \bar{B}) = 0.15$:

1. a) Determine whether 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 randomly chosen student plays soccer but not chess.
3. c) Calculate $P(\bar{F} \cup \bar{B})$ and interpret its meaning in the context of the problem.

A six-sided die is rolled that has been biased in such a way that the even faces all have the same probability of landing, and the odd faces also have the same probability as each other. 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 analysed. 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 analysed, what is the probability that exactly two present the mutation?
3. c) What is the minimum number of samples that should be analysed for the probability of finding at least one mutation to be greater than $0.95$?

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