Exercises: Definitions and properties

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".

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 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 ($P(T - P)$).

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

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 a secondary school, $25\%$ of the 1st-year Bachillerato students are signed up for guitar lessons, and $15\%$ are signed up for piano lessons. If we know that no student studies both instruments at the same time:

1. a) What are this type of events called in probability?

1. b) Calculate the probability that a randomly chosen student studies guitar or piano.

1. c) Calculate the probability that a student studies neither of these two instruments.

A card is drawn from a Spanish pack 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 picture card (jack, knight or king).
5. e) A card that is not an ace.

Hugo analyzes a survey carried out at his high school about the hobbies of Bachillerato students. According to his notes, the probability that a student plays video games regularly is $P(V) = 0,70$, and that they practice some team sport is $P(D) = 0,50$. Hugo publishes the following two statements in the school magazine:

1. "Events $V$ and $D$ are mutually exclusive because whoever spends a lot of time playing video games does not have time to train in a team."

2. "Therefore, the probability that a student plays video games or practices a team sport is $P(V \cup D) = 0,70 + 0,50 = 1,20$."

1. a) Identify the two conceptual errors Hugo has made and explain, with justification, which probability axioms or properties each of his statements violates.
2. b) Correct Hugo's analysis by deducing the minimum value that $P(V \cap D)$ must have for the data to be compatible with Kolmogorov's axioms.
3. c) If $P(V \cap D)$ takes exactly that minimum value, calculate the probability that a randomly chosen student only practices a team sport.

In a survey on sustainable mobility habits in a provincial capital, the probability that a citizen regularly uses public transport is $P(T) = 0,60$, uses a bicycle is $P(B) = 0,25$, and the probability that they use both means of transport in their daily life is $P(T \cap B) = 0,15$.

1. a) Before performing any mathematical operation, estimate with justification whether the probability that a person uses at least one of the two means of transport, $P(T \cup B)$, will be less than the direct sum $0,60 + 0,25 = 0,85$ and whether it can ever be less than $0,60$.
2. b) Apply the general formula for the probability of the union to calculate the exact value of $P(T \cup B)$ and check whether your initial estimate was correct.
3. c) From the previous result and using the complement rule together with De Morgan's laws, calculate the probability that a citizen uses neither of the two means of transport, $P(\overline{T} \cap \overline{B})$, and interpret whether the result obtained is consistent with the initial data.

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 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.

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:

	English ($A$)	French ($B$)	TOTAL
Level B1 ($M$)	$45$		$75$
Level B2 ($N$)		$20$
TOTAL	$100$

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?

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.

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

In a youth health centre, $30\ \%$ 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?

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.

In a charity raffle organised 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.

Based on data published by the National Statistics Institute (INE) in its survey on ICT equipment and usage in Spanish households, it is observed that $85\,\%$ of the adult population uses online banking (event $B$) and $65\,\%$ carries out procedures on public electronic portals (event $A$). Furthermore, it is found that the probability that a randomly selected citizen carries out neither of these two digital procedures is $0,08$.

1. a) Model the information formally by translating each percentage into probabilities of events expressed using set operations ($A$, $B$, unions, intersections, or complements).
2. b) Use the properties of the complement to obtain the numerical value of the probability that a citizen carries out at least one of the two procedures, $P(B \cup A)$.
3. c) Rigorously prove, from Kolmogorov's axioms, why events $B$ and $A$ cannot be mutually exclusive and calculate the exact percentage of citizens who carry out both procedures.
4. d) If a new event $D$ is defined: "downloading tax certificates", and we know that this procedure always requires accessing an electronic portal ($D \subseteq A$), reason using the monotonicity property whether a report claiming that $P(D) = 0,70$ would be mathematically permissible.

In an e-commerce company, a quality control check is conducted on shipments. The probability that a package experiences a delivery delay is $P(R) = 0,12$, the probability that it contains an incorrect item is $P(E) = 0,08$, and the probability that both issues occur simultaneously in the same order is $P(R \cap E) = 0,03$. The customer service department needs to calculate the probability of event $U$: "the order has exactly one of the two issues".

1. a) Carry out the calculation using Method 1: decompose $U$ as the union of two disjoint events using the difference of events ($R - E$ and $E - R$), applying the corresponding probability properties.
2. b) Carry out the calculation using Method 2: express the probability of $U$ from the difference between the union and the intersection, $P(R \cup E) - P(R \cap E)$, first justifying with logical reasoning why this expression is equivalent to $U$.
3. c) Compare both approaches: verify that they lead to the same value and justify which of the two methods would be more advantageous if an external audit directly gave you the global values $P(R \cup E) = 0,17$ and $P(R \cap E) = 0,03$ without providing you with $P(R)$ or $P(E)$ separately.

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 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 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.

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.

In a school, $65\%$ of the students play football ($F$) 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 football but not chess.
3. c) Calculate $P(\bar{F} \cup \bar{B})$ and interpret its meaning in the context of the problem.

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?

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 $0,95$?

An investment management department evaluates two unfavorable events, $A$ and $B$, that could affect an asset portfolio. Management establishes as a safety constraint that the probability of neither event occurring must be at least $0,60$ ($P(\overline{A} \cap \overline{B}) \geq 0,60$). It is known with certainty that the probability of event $A$ is $P(A) = 0,35$.

1. a) Assuming that events $A$ and $B$ were mutually exclusive, determine the maximum value that the probability $P(B)$ could reach in order to respect management's safety criterion.
2. b) If a new expert report concludes that $P(B) = 0,25$, deduce the inequality that $P(A \cap B)$ must satisfy to guarantee that $P(\overline{A} \cap \overline{B}) \geq 0,60$. Next, determine the complete closed interval $[k_{\min}, k_{\max}]$ of admissible values for $P(A \cap B)$, taking into account Fréchet's axiomatic bounds.
3. c) Design an explicit probability assignment for the partition formed by the four mutually exclusive elementary events $\{A \cap B,\ A \cap \overline{B},\ \overline{A} \cap B,\ \overline{A} \cap \overline{B}\}$ such that $P(A) = 0,35$, $P(B) = 0,25$, and $P(\overline{A} \cap \overline{B}) = 0,65$ hold simultaneously, checking that all of Kolmogorov's axioms are satisfied.

The State Meteorological Agency (AEMET) analyzes the frequency of two summer weather warnings in an agricultural region: heatwave warning ($C$) and dry thunderstorm warning ($T$). Based on the historical data of recent years, it is estimated that $P(C) = 0,75$ and $P(T) = 0,40$.

1. a) Deduce step by step, using exclusively Kolmogorov's axioms and the properties of addition, the closed numerical interval $[L_{\min}, L_{\max}]$ within which the probability that both warnings coincide, $P(C \cap T)$, must necessarily lie, justifying analytically where each of the two bounds comes from.
2. b) A farmers' association presents a report claiming that the probability of having a calm day with neither warning is $30\,\%$ ($P(\overline{C} \cap \overline{T}) = 0,30$). Mathematically prove that this statement is untenable and false, analyzing the contradiction that arises both when calculating $P(C \cap T)$ and when applying the monotonicity property to event $C$ with respect to the union $C \cup T$.
3. c) If a third warning for hail is added ($G$) and it is known that every hail episode is necessarily accompanied by a dry thunderstorm ($G \subseteq T$), prove from the additivity axiom that $P(T \cap \overline{G}) = P(T) - P(G)$. Calculate this probability knowing that $P(G) = 0,15$ and interpret its meaning for the agricultural insurance sector.