Exercises: Events

A 2nd-year Bachillerato student is organizing their digital folders of notes. In their "Economía" folder they have 12 PDF files, 8 text documents and 5 diagrams in image format. If they pick a file at random to review before an exam, determine:

1. a) The sample space of the experiment.

1. b) The probability that the chosen file is a diagram in image format.

1. c) The probability that the chosen file is not a text document.

In a Year 13 class, a survey on social media use was conducted. Let event $A$ be "using Instagram" and event $B$ be "using TikTok". Define the following situations using set operations on events (union, intersection or complement):

1. a) The student uses both social networks.
2. b) The student uses at least one of the two networks.
3. c) The student does not use Instagram.
4. d) The student uses Instagram but not TikTok.

Consider two events $A$ and $B$ associated with a random experiment. If we know that $P(A) = 0,6$, $P(B) = 0,4$ and $P(A \cap B) = 0,2$, calculate the following probabilities:

1. a) $P(A \cup B)$

1. b) $P(\bar{A})$ (probability of the complement of event $A$).

1. c) $P(\bar{A} \cap \bar{B})$ using De Morgan's Laws.

The Venn diagram shows only the data given in the statement.

An esports team has players of three nationalities on its roster: 5 Spanish, 3 French and 2 Italian. If a captain is randomly selected for the next tournament, calculate:

1. a) The probability that they are French.
2. b) The probability that they are not Spanish.
3. c) Are the events "being Spanish" and "being Italian" compatible or incompatible? Justify your answer.

Given two events $M$ and $N$ that are incompatible with each other, where $P(M) = 0,35$ and $P(N) = 0,45$, find the probability that:

1. a) Event $M$ or event $N$ occurs.
2. b) Neither of the two events occurs.
3. c) $M$ and $N$ occur simultaneously.

On a streaming platform, $70\%$ of users watch action series ($S$), $40\%$ watch documentaries ($D$) and $25\%$ watch both types of content. Calculate the probability that a randomly chosen user:

1. a) Watches action series or documentaries.

1. b) Watches neither of these two types of content.

Let $A$ and $B$ be two events such that $P(A) = 0,5$, $P(\bar{B}) = 0,7$ and $P(A \cup B) = 0,65$. Calculate the probability of the intersection $P(A \cap B)$ and determine whether the events are compatible.

A ball is drawn from an urn containing 10 balls numbered 1 to 10. Consider the events:

$A = \{ \text{drawing an even number} \}$

$B = \{ \text{drawing a multiple of 3} \}$

1. a) Write down the elements that make up the events $A \cup B$ and $A \cap B$.
2. b) Calculate $P(A \cup B)$ using Laplace's rule on the count of their elements.
3. c) Verify the previous result by applying the formula $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.

In a secondary school, the guidance department is analyzing enrollment options in the 2nd year of Bachillerato regarding two elective subjects: French ($F$) and Information Technology ($T$). It is known that $30\,\%$ of the students take French, $45\,\%$ take Information Technology, and $15\,\%$ take both.

Alejandro, to calculate the probability that a randomly chosen student takes neither of the two subjects, writes on his exam:

1. a) Identify the conceptual error Alejandro made with De Morgan's laws or operations with events, and explain in a reasoned way why his formula is not correct.
2. b) Correct the procedure and calculate the true value of $P(\overline{F} \cap \overline{T})$.
3. c) Based on the data, justify whether events $F$ and $T$ are compatible or mutually exclusive, and whether they are complementary events.

Based on data from the Survey on Equipment and Use of Information and Communication Technologies in Households by the INE, it is estimated that among young people aged 16 to 24 in Spain, $92\,\%$ use social networks daily (event $R$) and $68\,\%$ access digital banking platforms or mobile wallets (event $B$). Furthermore, $65\,\%$ claim to perform both actions daily.

If a person from this group is chosen at random:

1. a) Express formally using event operations ($R$, $B$, and their complements) each of the following situations:

i) Uses social networks daily but not digital banking.

ii) Performs neither of the two activities daily.

iii) Performs at least one of the two activities daily.

1. b) Calculate the probability of each of the three events in the previous section.
2. c) Formally justify whether the events "using social networks daily" and "using digital banking daily" are complementary events, stating the two conditions they would have to satisfy to be so.

Let $A$ and $B$ be two events in the same random experiment for which the following probabilities are known: $P(A) = 0,6$, $P(B) = 0,5$ and $P(\bar{A} \cap \bar{B}) = 0,2$.

Based on the properties of probability and De Morgan's laws, calculate:

1. a) The probability of the union of the events, $P(A \cup B)$.
2. b) The probability of the intersection, $P(A \cap B)$.
3. c) State with reasons whether the events $A$ and $B$ are compatible or incompatible.

In a study on digital content consumption among 16-year-olds, it was found that $70 \%$ use the platform StreamVibe and $45 \%$ use CloudMusic. Furthermore, it is known that $25 \%$ of young people do not use either of the two platforms. If a young person is chosen at random, calculate:

1. a) The probability that they use both platforms.
2. b) The probability that they use StreamVibe only.
3. c) The probability that they use CloudMusic but not StreamVibe.

We have the following partial contingency table showing the distribution of a group of 120 Sixth Form students according to their participation in extracurricular activities (Sport or Music):

1. a) Complete the contingency table in your notebook.
2. b) Calculate the probability that a randomly chosen student does not participate in either activity.
3. c) Are the events "Participating in Music" and "Participating in Sport" independent? Justify your answer mathematically.

A Year 13 student has prepared 18 of the 25 topics that make up the Geography syllabus. In the final exam, the teacher chooses two topics at random (without replacement). Calculate:

1. a) The probability that the student knows both topics.
2. b) The probability that they know neither.
3. c) The probability that they know at least one of the two selected topics.

Let $A$ and $B$ be two events such that $P(A) = 0,4$ and $P(B) = 0,3$. Calculate the probability of the union $P(A \cup B)$ under each of the following assumptions:

1. a) $A$ and $B$ are incompatible events.
2. b) $A$ and $B$ are independent events.
3. c) The event $B$ is contained in the event $A$ ($B \subset A$).

In a role-playing game, a player must pass two consecutive challenges. The probability of passing the first challenge is $0,7$. If they pass the first, the probability of passing the second is $0,8$, but if they fail the first, the probability of passing the second drops to $0,4$.

1. a) Draw the tree diagram representing this situation.
2. b) Calculate the probability that the player passes both challenges.
3. c) Calculate the probability that the player passes only one of the two challenges.

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

From a Spanish deck of 40 cards, two cards are drawn successively without replacing the first card in the deck. Determine the following probabilities:

1. a) Both cards are face cards (Jack, Knight or King).
2. b) The first card is an Ace and the second is a King.
3. c) At least one of the two cards is Coins.

Consider the events $A$ and $B$ such that $P(A) = 0,5$, $P(B|A) = 0,4$ and $P(A \cup B) = 0,65$. From these data:

1. a) Find the probability of the intersection $P(A \cap B)$.
2. b) Calculate the value of $P(B)$.
3. c) Determine, giving reasons, whether the events $A$ and $B$ are independent.

According to climatological records from AEMET for a capital city in northern Spain in November, the probability of it raining on any given day is $P(L) = 0{,}70$ and the probability of recording strong wind gusts is $P(V) = 0{,}50$. Likewise, the probability that a day has neither rain nor strong wind is $P(\overline{L} \cap \overline{V}) = 0{,}10$.

1. a) Before calculating any exact value, make a reasoned estimate: can the probability of it raining and being windy at the same time, $P(L \cap V)$, be less than $0{,}20$? Justify your estimate by deducing a mandatory lower bound from the sum $P(L) + P(V)$ and the fact that a probability can never exceed $1$.
2. b) Using De Morgan's laws and the properties of the addition of probabilities, calculate the exact value of the union $P(L \cup V)$ and the intersection $P(L \cap V)$.
3. c) Check whether the result obtained confirms your previous estimate and calculate the probability that on a day in November exactly one of the two meteorological phenomena occurs (rain without strong wind, or strong wind without rain).

A fleet of e-commerce delivery vans operates in the metropolitan area of Valencia. On a working day, the probability of a vehicle experiencing a mechanical breakdown is $P(M) = 0{,}18$ and that of experiencing a severe delay due to traffic congestion is $P(T) = 0{,}32$. It is also known that the probability of experiencing at least one of these two problems is $P(M \cup T) = 0{,}40$.

The goal is to calculate the probability that a vehicle finishes the day experiencing at most one of the two incidents.

1. a) Carry out the calculation following Method 1: define the requested event as the complement of experiencing both problems simultaneously, applying the relationship with the complementary event.
2. b) Carry out the calculation following Method 2: decompose the event into the union of three mutually exclusive events (breakdown only, traffic only, or neither of the two incidents) and sum their probabilities using the difference of events and De Morgan's laws.
3. c) Compare both methods: check that they yield the same result and argue which of the two paths you consider more efficient and reliable for a university entrance exam, justifying your choice.

On a streaming platform, it has been observed that $65\%$ of users under 20 watch animated series (event $A$) and $40\%$ watch nature documentaries (event $D$). Furthermore, it is known that the probability that a user does not watch animated series given that they do not watch nature documentaries is $0,5$.

Give reasoned answers to the following:

1. a) The probability that a user watches both types of content.
2. b) The probability that a user watches animated series but not nature documentaries.
3. c) Are the events $A$ and $D$ independent? Justify your answer using the previous results.

A video game development team is analysing the bugs in their latest release. They have classified the bugs into three categories: Critical ($C$), Graphics ($G$) and Sound ($S$). Following a mass test, the following data were obtained:

• $10\%$ of the bugs belong to all three types at the same time.

• The probability that a bug is Critical is $P(C) = 0,4$.

• The probability that it is Graphics is $P(G) = 0,5$.

• The probability that it is Sound is $P(S) = 0,35$.

• The probability that a bug is Critical and Graphics is $0,2$.

• The probability that it is Graphics and Sound is $0,15$.

• The probability that a bug does not belong to any of these three categories is $0,2$.

Calculate the probability that a randomly selected bug is:

1. a) Critical and Sound.
2. b) Exclusively a Graphics bug.
3. c) Sound, given that it is neither Critical nor Graphics.

Let $A$ and $B$ be two events in a random experiment such that their probabilities are $P(A) = x$ and $P(B) = y$. It is known that the union event has a probability of $P(A \cup B) = 0,75$ and that the intersection event satisfies $P(A \cap B) = 0,15$.

If it is also known that the probability of $A$ occurring given that $B$ does not occur is $P(A | \bar{B}) = 0,5$, determine the values of $x$ and $y$. Are $A$ and $B$ compatible events? Are they independent?

In an esports championship, the plays of a pro player are analysed. The events $K$ (getting a kill) and $W$ (winning the round) are defined. The analyst determines that:

• The probability of winning the round having achieved a kill is $P(W|K) = 0,8$.

• The probability of winning the round without having achieved any kills is $P(W|\bar{K}) = 0,3$.

• The probability that in a round the player wins or gets a kill is $0,85$.

Using event algebra, calculate:

1. a) The probability that the player gets a kill in a round, $P(K)$.
2. b) The probability that the player wins the round and does not get any kills.
3. c) Show whether the events "winning the round" and "getting a kill" are dependent or independent.

Using the properties of probability and De Morgan's laws, prove whether the following statement is true or false:

"If $A$ and $B$ are two incompatible events with non-zero probabilities, then $P(\bar{A} | B) = 1$ always holds".

Support your reasoning with an original numerical example where $P(A) = 0,3$ and $P(B) = 0,4$.

A cybersecurity system has two independent protective barriers, $B_1$ and $B_2$. The probability that an external attack bypasses the first barrier is $P(B_1) = 0,15$ and the probability that it bypasses the second is $P(B_2) = 0,10$.

Calculate the probability that an attack:

1. a) Bypasses at least one of the two barriers.
2. b) Bypasses exactly one of the barriers.
3. c) Is not detected by either barrier, knowing that the attack has managed to bypass the first protection.

In a study on social media use, two 2nd year of Bachillerato students are selected consecutively and without replacement. It is known that $60\%$ of the school's students use the social network X.

If the probability that the second selected student uses the social network X, knowing that the first one uses it, is $35/59$, determine the total number of 2nd year Bachillerato students in that school.

Consider three events $A$, $B$ and $C$ such that $A$ and $B$ are independent, and $A$ and $C$ are mutually exclusive. The following probabilities are known:

$P(A) = 0,4$; $P(B) = 0,5$; $P(C) = 0,2$ and $P(B \cap C) = 0,1$.

Find the probability of the following compound events:

1. a) $P(A \cup B \cup C)$

1. b) $P((A \cap B) \cup C)$

1. c) $P(A \mid \bar{C})$

A financial analyst designs a model to evaluate two credit risk factors in SMEs: factor $A$ (immediate lack of liquidity) and factor $B$ (sharp drop in sales). The initial model requires two fixed conditions to be met:

• The probability that a company presents neither of the two risk factors is $P(\overline{A} \cap \overline{B}) = 0{,}25$.

• The probability of presenting only factor $A$ is twice the probability of presenting only factor $B$; that is, $P(A \cap \overline{B}) = 2 \cdot P(\overline{A} \cap B)$.

1. a) If preliminary data set the probability of both factors occurring at the same time at $P(A \cap B) = 0{,}15$, reconstruct with justification the individual values of $P(A)$ and $P(B)$.
2. b) Formulate and solve the following inverse problem: determine what values $P(A)$ and $P(B)$ should take if the restriction that risk factors $A$ and $B$ are completely mutually exclusive is imposed, while maintaining the two initial conditions of the model.
3. c) Prove whether it is possible or impossible for $P(A \cap B) = 0{,}80$ under the initial conditions of the problem and deduce the maximum value that $P(A \cap B)$ could theoretically reach. Justify your answer in terms of event inclusion.

A report from a social observatory on responsible consumption among young people aged 18 to 25 in Spain yields the following sample data:

• $70\,\%$ regularly purchase second-hand or ethically manufactured clothing (event $S$).

• $55\,\%$ regularly consume organic or local food (event $E$).

During an interview, the organization's spokesperson states to the media: "According to our calculations, at least $40\,\%$ of young people do not engage in either of these two sustainable practices."

1. a) Translate the statements from the report and the spokesperson's declarations into the formal language of events and probabilities.
2. b) Rigorously applying De Morgan's laws and the axioms of probability, prove whether the spokesperson's statement is mathematically consistent with the report's data or if it contains an insurmountable contradiction.
3. c) Assuming the report's percentages are true ($P(S) = 0{,}70$ and $P(E) = 0{,}55$), determine the exact closed interval $[p_{\min}, p_{\max}]$ in which the probability of performing both practices at the same time, $P(S \cap E)$, must lie for the model to be mathematically consistent. Explain the meaning of each of the endpoints of the interval regarding the inclusion or mutual relationship between both events.