Exercises: Events

In the study of everyday phenomena, it is essential to distinguish between certainty and chance. Classify the following experiments as random or deterministic:

1. a) Taking a penalty in a training session and observing whether the ball goes into the goal.
2. b) Measuring the time taken for an object to fall from a height of 5 metres in a vacuum.
3. c) Observing the number of WhatsApp messages you receive between 17:00 and 18:00 on any given Monday.
4. d) Calculating the total cost of buying 3 kilograms of oranges if the price per kilogram is $1,25 \text{ €}$.
5. e) Drawing a card from a Spanish pack of cards and observing whether it is clubs.

We have a wheel of fortune divided into 8 equal sectors, numbered from 1 to 8. The wheel is spun and the number indicated by the arrow is recorded.

Based on this experiment:

1. a) Write down the sample space $E$.
2. b) State the outcomes that form event $A = \text{"getting a number greater than 5"}$.
3. c) State the outcomes that form event $B = \text{"getting an even number"}$.
4. d) Describe the complementary event $\bar{A}$ in words.

Two cubic (six-sided) dice are rolled and the sum of the numbers obtained is recorded.

1. a) What is the sample space $E$ of this experiment?
2. b) Write down the elementary events that make up event $S = \text{"the sum is a multiple of 5"}$.
3. c) Determine whether event $I = \text{"the sum is a number less than 2"}$ is a possible, impossible or certain event.

In a bag there are 10 balls numbered from 1 to 10. We consider the following events:

$A = \{1, 3, 5, 7, 9\}$

$B = \{6, 7, 8, 9, 10\}$

Find the elements of the following resulting events:

1. a) $A \cup B$
2. b) $A \cap B$
3. c) $\bar{A}$ (complementary event of $A$)
4. d) $A - B$ (outcomes that are in $A$ but not in $B$)

Consider the experiment of drawing a card from a Spanish deck of 40 cards. State whether the following pairs of events are compatible or incompatible:

1. a) $A = \text{"Drawing a King"}$ and $B = \text{"Drawing a Jack"}$.
2. b) $C = \text{"Drawing a Gold"}$ and $D = \text{"Drawing a Face Card"}$.
3. c) $E = \text{"Drawing an even number"}$ and $F = \text{"Drawing the Ace of Swords"}$.
4. d) $G = \text{"Drawing a Knight"}$ and $H = \text{"Drawing a card of Cups"}$.

A group of friends analyse their hobbies. Let $D$ be the event "playing sport" and $M$ be the event "playing a musical instrument". Express the following situations symbolically using operations on events:

1. a) The friend plays sport or plays an instrument (or both).
2. b) The friend plays sport and also plays an instrument.
3. c) The friend plays sport but does not play any instrument.
4. d) The friend neither plays sport nor plays any instrument.

Observe the following Venn diagram representing events $A$ and $B$ within a sample space $E$.

If point $x$ represents an outcome of the experiment, state whether the following statements are true or false based on the graph:

1. a) The outcome $x$ belongs to event $A \cup B$.
2. b) The outcome $x$ belongs to event $A \cap B$.
3. c) Events $A$ and $B$ are incompatible.
4. d) The outcome $x$ belongs to event $A - B$.

In a Year 12 class, there are 12 students who have chosen French as a second language, 15 who have chosen German and 3 who have chosen both. If we choose a student at random:

1. a) What is the total number of students who study at least one of the two languages (union event)?
2. b) How many students study only German?
3. c) Using Laplace's rule, if there are 30 students in total in the class, calculate the probability that the chosen student studies German.

In a high school, a study on school commute habits is conducted. Event $B$ is defined as "the student comes by bicycle" and event $C$ as "the student comes by private car". In a class debate, Marta states emphatically: "Since a student cannot arrive simultaneously by bicycle and by car on the same trip, events $B$ and $C$ are complementary; therefore, it is necessarily true that $P(B) + P(C) = 1$".

1. a) Identify the conceptual error Marta makes by confusing mutually exclusive events with complementary events.
2. b) Explain the theoretical difference between both concepts using the definition of sample space $E$.
3. c) Describe in words what the event $\overline{B \cup C}$ represents in this context and give an example of an outcome of the experiment that belongs to this event to prove to Marta that their sum does not have to be $1$.

Consider the random experiment consisting of rolling a fair six-sided die with faces numbered from 1 to 6, whose sample space is $E = \{1, 2, 3, 4, 5, 6\}$. Design and write down the sets of outcomes corresponding to two compound events, $A$ and $B$, non-empty and different from $E$, that simultaneously satisfy the following three conditions:

1) $A$ and $B$ are compatible ($A \cap B \neq \emptyset$).

2) The difference $A - B$ is an elementary event (contains a single outcome).

3) The union event $A \cup B$ does not contain any prime number (remember that the prime numbers on the die are 2, 3, and 5).

Explicitly state the elements of $A$, $B$, $A \cap B$, $A - B$, and $A \cup B$, and justify step by step why your choice satisfies each of the required conditions.

In a survey carried out among 1st-year Bachillerato students about their subscriptions to streaming platforms, it was found that $60\%$ are subscribed to Netflix ($N$), $45\%$ to HBO Max ($H$) and $20\%$ are not subscribed to either of the two. If we choose a student at random, calculate:

1. a) The probability that they are subscribed to both platforms.

1. b) The probability that they are subscribed to Netflix but not to HBO Max ($P(N - H)$).

1. c) Represent the situation using a Venn diagram (draw the sample space $E$ and the sets $N$ and $H$), indicating the probabilities of each region.

Given two events $A$ and $B$ in a random experiment, we know that the probability that neither of them occurs is $P(\bar{A} \cap \bar{B}) = 0,3$. If it is also known that $P(A) = 0,4$ and $P(B) = 0,5$, determine:

1. a) Are $A$ and $B$ incompatible events? Justify your answer.
2. b) Are $A$ and $B$ independent events?
3. c) Calculate $P(A \cap \bar{B})$.

In an e-sports tournament, $70\%$ of the participants use a mechanical keyboard ($T$) and $40\%$ use an ergonomic mouse ($R$). We know that the probability that a randomly chosen participant uses neither a mechanical keyboard nor an ergonomic mouse is $0,15$.

1. a) Calculate the probability that a player uses both.
2. b) If we know that a player uses a mechanical keyboard, what is the probability that they also use an ergonomic mouse?
3. c) Is the use of an ergonomic mouse independent of the use of a mechanical keyboard?

Observe the following Venn diagram representing the probabilities of events $A$ (studying French) and $B$ (studying German) in a language centre.

Using the data from the chart, calculate:

1. a) $P(A \cup B)$
2. b) $P(\bar{A} | B)$
3. c) $P(\bar{A} \cap \bar{B})$ using De Morgan's Laws.

A data analyst studies the probability of an Instagram post being successful. They define the events $V$ (the post is a video) and $L$ (the post receives more than 500 likes). It is known that $P(V) = 0,25$, $P(L) = 0,40$ and $P(V \cup L) = 0,55$.

1. a) Calculate $P(V \cap L)$.
2. b) Determine whether the events "being a video" and "receiving many likes" are independent.
3. c) Calculate the probability that the post receives many likes given that it is not a video.

In a safe there are two compartments. Compartment A contains 4 10€ notes and 6 20€ notes. Compartment B contains 7 10€ notes and 3 20€ notes. A die is rolled: if a number less than 3 is obtained, compartment A is chosen; otherwise, compartment B is chosen. Then, a note is drawn at random.

1. a) Draw the corresponding tree diagram.
2. b) What is the probability that the note drawn is a 20€ note?
3. c) If the note drawn is a 10€ note, what is the probability that it came from compartment A?

You can do it on paper and send a photo with your answer.

A Year 12 project team consists of 5 girls and 3 boys. The teacher must choose 2 people at random to present the project. Calculate:

1. a) The sample space of the experiment (considering the gender of those chosen).
2. b) The probability that both people chosen are of the same gender.
3. c) The probability that at least one of those chosen is a girl.

A study has been carried out on the use of sunglasses and eye strain in drivers. Out of 200 drivers, 120 wear sunglasses regularly. Of those who wear sunglasses, $10\%$ state that they suffer from eye strain after a long journey, whereas among those who do not wear them, the percentage suffering from eye strain rises to $35\%$.

1. a) Organise the information in a contingency table.
2. b) If we choose a driver at random, what is the probability that they suffer from eye strain?
3. c) If a driver states that they do not suffer from eye strain, what is the probability that they wear sunglasses?

According to a report on connectivity and digital entertainment in households of an autonomous community, $80\,\%$ of families have a high-speed fiber-optic connection ($F$) and $60\,\%$ have a subscription to an on-demand television platform ($T$). Furthermore, $50\,\%$ of households have both services simultaneously ($F \cap T$).

1. a) Before carrying out any formal calculation, reasonably estimate whether the probability that a randomly chosen household has neither of the two services ($P(\overline{F} \cap \overline{T})$) will be greater than, less than, or equal to $0,15$. Briefly explain the intuition or balance of percentages on which you base your answer.
2. b) Express the event "having neither of the two services" using event operations with De Morgan's laws and calculate its exact probability by applying the corresponding axioms.
3. c) Compare the value obtained with your initial estimate, calculate the probability that a household has fiber but not on-demand television ($P(F - T)$), and reason whether the proportion of households that have only one of the two services exceeds that of households having neither.

From data of the Active Population Survey (EPA) for young people aged 18 to 24 in a Spanish province, two events are modeled when selecting a young person at random: $T$ ("having paid employment") and $E$ ("enrolled in formal education"). The records indicate that $P(T) = 0,38$, $P(E) = 0,52$, and the probability that a young person engages in at least one of the two activities is $P(T \cup E) = 0,78$.

1. a) Translate each of the following everyday situations into the formal notation of event operations ($T$, $E$, unions, intersections, or complements):

i) The young person combines work and academic education.

ii) The young person does neither of the two activities (a situation known as "NEET").

iii) The young person works, but does not study.

1. b) Calculate the probability of each of the three previous events by rigorously applying the properties of operations (union formula, De Morgan's laws, and difference event).
2. c) Justify whether events $T$ and $E$ are compatible or mutually exclusive and interpret the socioeconomic meaning of $T \cap E \neq \emptyset$ in this age group.

In a study on digital media consumption among teenagers, two events are analysed: $A$ (the user regularly uses a microblogging platform) and $B$ (the user consumes content on video streaming platforms). After conducting a survey, the following probability data have been obtained:

• The probability that a user does not use either of the two platforms is $P(\bar{A} \cap \bar{B}) = 0,15$.

• The probability that they regularly use the microblogging platform is $P(A) = 0,6$.

• The probability that they do not use the streaming platform given that they do use the microblogging platform is $P(\bar{B}/A) = 0,4$.

Calculate, giving reasons, the following probabilities:

1. a) The probability that the user consumes streaming content, $P(B)$.
2. b) The probability that they use the microblogging platform but not the streaming platform, $P(A - B)$.
3. c) Are the events $A$ and $B$ independent? Justify your answer using calculations.

Consider a random experiment where the sample space is $E$. Let $S_1$ and $S_2$ be two events such that their probabilities depend on an unknown value $k \in \mathbb{R}$. It is known that:

• $P(S_1) = k$

• $P(S_2) = k^2 + 0,1$

• $P(S_1 \cup S_2) = 0,7$

Determine the value of $k$ knowing that the events $S_1$ and $S_2$ are incompatible. Once $k$ has been found, check whether it would be possible for $S_1$ and $S_2$ to be independent under the same individual probability conditions (ignoring the previous union). Justify whether there exists any value of $k$ that allows independence and the union simultaneously.

A cybersecurity company has installed two intrusion detection systems, $D_1$ and $D_2$, which work independently. The probability that, in the event of an external attack, system $D_1$ raises the alarm is $0,92$. On the other hand, the probability that system $D_2$ does not detect the attack is $0,15$.

The diagram shows the possible branches (✓ = detects, ✗ = does not detect).

If an unexpected attack occurs, calculate:

1. a) The probability that at least one of the two systems detects the attack.

1. b) The probability that only system $D_2$ raises the alarm.

1. c) Given that the attack has been detected by at least one of the systems, what is the probability that both systems worked correctly?

Let $A$, $B$ and $C$ be three events in the same random experiment such that:

• $A$ and $B$ are independent.

• $A$ and $C$ are incompatible.

• $P(A) = 0,4$; $P(B) = 0,5$ and $P(C) = 0,2$.

Calculate in detail, applying the properties of probability:

1. a) $P(A \cup B)$
2. b) $P(A \cup C)$
3. c) $P(\bar{A} \cap \bar{B})$
4. d) $P((A \cup B) \cap C)$ knowing also that $B$ and $C$ are independent.

In a prestigious music academy, $60\%$ of the students play the piano ($P$), $45\%$ play the violin ($V$) and $20\%$ play neither of these two instruments. A student is selected at random.

1. a) Determine the probability that they play both instruments.
2. b) If we know that the selected student plays the violin, what is the probability that they do not play the piano?
3. c) Three students are chosen at random independently. Calculate the probability that at least one of them plays the piano but not the violin.

Reasonably prove, using De Morgan's laws and the axiomatic properties of probability, the following equality for any two events $A$ and $B$:

$P(\bar{A} \cap \bar{B}) - P(\bar{A}) = P(A) - P(A \cup B)$

Subsequently, interpret what this equality means in terms of the relationship between events $A$ and $B$ when the result is equal to zero.

A diagnostic laboratory analyzes water samples to detect a specific bacterium. It is known that $5\%$ of the springs in a region are contaminated ($C$). The test used has a probability of giving a false positive (testing positive when there is no bacterium) of $2\%$, and a probability of giving a false negative (testing negative when there is bacterium) of $1\%$.

1. a) Construct the tree diagram corresponding to this situation.

1. b) Calculate the probability that the test result is positive ($+$).

1. c) If the test of a spring has come out positive, what is the probability that it is actually contaminated? Comment on the result obtained from the point of view of the reliability of the test.

You can do it on paper and send a photo with your answer.

In an e-sports tournament, a team has to pass three consecutive knockout stages to win the prize. The probabilities of passing each stage depend on success in the previous stage due to accumulated fatigue:

• The probability of passing Stage 1 is $P(F_1) = 0,8$.

• If they pass Stage 1, the probability of passing Stage 2 is $P(F_2/F_1) = 0,7$.

• If they do not pass Stage 1, they are automatically eliminated (obviously there is no Stage 2).

• If they have passed the first two stages, the probability of winning Stage 3 is $P(F_3 / F_1 \cap F_2) = 0,5$.

Calculate:

1. a) The probability that the team is eliminated in exactly Stage 2.
2. b) The probability that the team wins the tournament (passes all three stages).
3. c) Given that the team did not win the tournament, what is the probability that they did not even pass the first stage?

A Spanish financial institution audits electronic payments to detect suspicious transactions using two types of automatic alerts: inconsistent geographic location ($L$) and unusually high amount ($I$). The probabilities $P(L) = 0,15$ and $P(I) = 0,10$ have been determined, and it is known that the probability that a transaction triggers neither of the two alerts is $P(\overline{L} \cap \overline{I}) = 0,80$. The technical team needs to find the probability of event $D$: "the transaction triggers exactly one of the two alerts".

To calculate $P(D)$, two analysts propose different procedures:

• Method 1 (Disjoint decomposition): Obtain $P(L \cap I)$ using De Morgan's laws; calculate separately the probabilities of the disjoint events $L - I$ and $I - L$, and add them: $P(D) = P(L - I) + P(I - L)$.

• Method 2 (Global subtraction): Determine $P(L \cup I)$ from the complement and directly subtract the intersection: $P(D) = P(L \cup I) - P(L \cap I)$.

1. a) Develop both methods step by step, showing intermediate operations, and verify that both yield the same probability value.
2. b) Formally prove, using the definitions of set operations (union, intersection, and difference), that $(L - I) \cup (I - L) = (L \cup I) - (L \cap I)$.
3. c) Compare both approaches and justify which one is more algebraically efficient and which one minimizes the risk of forgetting the overlap of events if the data comes from a real database.

In a technology consulting firm, candidates for a technical role are evaluated based on two skills: cloud computing certification ($C$) and proficiency in functional programming ($F$). From previous evaluations, it is known that $75\,\%$ hold the cloud certification ($P(C) = 0,75$) and $40\,\%$ are proficient in functional programming ($P(F) = 0,40$). Initially, the joint figure $P(C \cap F)$ is not available.

1. a) Reasonably determine the closed interval $[k_{\min}, k_{\max}]$ of admissible values for the probability that a randomly selected candidate possesses both skills, $P(C \cap F)$. Justify the lower bound from the property $P(C \cup F) \le 1$ and the upper bound from the inclusion relation $(C \cap F) \subseteq F$.
2. b) Inverse problem: Management sets as a selection criterion that the probability of rejecting a candidate for possessing neither of the two skills, $P(\overline{C} \cap \overline{F})$, must be at least $0,10$. Prove what restriction this condition imposes on $P(C \cap F)$ and determine the new valid range for said probability under this requirement.
3. c) Based on the previous results, rigorously prove why it is mathematically impossible for events $C$ and $F$ to be mutually exclusive in this company, resorting to Kolmogorov's axioms and the sample space.