Exercises: Properties of probability

In a study on mobile app usage among teenagers, it has been determined that the probability of a student having a video editing app installed is $P(V) = 0,35$. Calculate the probability that a student chosen at random does not have this app installed.

For the graduation trip, Year 13 students must choose between three possible destinations: Mojácar, Mallorca or Tenerife. It is known that the probability of them choosing Mojácar is $P(M) = 0,42$ and that of choosing Mallorca is $P(L) = 0,38$. Given that the events are mutually exclusive (they can only choose one destination), calculate:

1. a) The probability that the students travel to Mojácar or Mallorca.
2. b) The probability that they choose neither of these two destinations and go to Tenerife.

According to a school's statistics, $75\,\%$ of the students regularly use Instagram ($I$), $60\,\%$ use TikTok ($T$) and $45\,\%$ use both social networks at the same time.

Calculate the probability that a randomly selected student uses Instagram or TikTok, that is, $P(I \cup T)$.

A video game developer is analysing the probability of obtaining rare items from a loot box. The possible outcomes are: "Common Item" ($C$), "Rare Item" ($R$) and "Epic Item" ($E$). State whether the following probability assignment is valid according to Kolmogorov's axioms and justify your answer:

$P(C) = 0,65$

$P(R) = 0,20$

$P(E) = 0,25$

Consider events $A$ and $B$ associated with a random experiment. If we know that $P(A) = 0,5$, $P(B) = 0,4$ and that the probability of neither $A$ nor $B$ occurring is $P(\bar{A} \cap \bar{B}) = 0,3$, apply de Morgan's Laws to find the probability of the union, $P(A \cup B)$.

In a box of materials for a technology project there are 50 resistors. It is known that some are $100\ \Omega$ ($A$) and others are $220\ \Omega$ ($B$), and some are new and others used. One resistor is drawn at random. If $P(A) = 0,6$ and events $A$ and $B$ are complementary, determine:

1. a) The probability of drawing a $220\ \Omega$ resistor.
2. b) What is the value of $P(A \cap B)$? Justify your answer based on the relationship between complementary events.

Let $X$ and $Y$ be two events such that $P(X) = 0,25$ and $P(Y) = 0,55$. Given that these events are mutually exclusive, calculate the probability that:

1. a) Event $X$ or event $Y$ occurs.
2. b) Event $X$ does not occur.
3. c) $X$ and $Y$ occur at the same time.

From the data represented in the following Venn diagram, where the values indicate the probabilities of each region, calculate the required probabilities:

1. a) $P(A)$
2. b) $P(B)$
3. c) $P(A \cup B)$

The State Meteorological Agency (AEMET) forecasts for a weekend in Santander a probability of rain on Saturday of $P(S) = 0,70$, a probability of rain on Sunday of $P(D) = 0,60$, and a probability of rain on both days of $P(S \cap D) = 0,50$.

1. a) Before carrying out any formal numerical calculation, make a reasoned estimate of whether the probability of rain on at least one of the two days ($P(S \cup D)$) must be greater than, equal to, or less than $0,70$. Justify your prediction using the inclusion relation between the events $S$ and $S \cup D$.

1. b) Calculate the exact value of $P(S \cup D)$ using the property of the union of compatible events and check whether your initial estimate was correct.

1. c) Using De Morgan's laws, find the probability that the weekend will be completely dry ($P(\overline{S} \cap \overline{D})$) and assess whether this result is consistent with the initial data.

In a secondary school, a survey on reading habits is carried out. Let event $L$ be defined as: "the student regularly reads books in paper format" with $P(L) = 0,60$, and event $D$ as: "the student reads in digital format" with $P(D) = 0,55$. Lucía, a 2nd year Bachillerato student, states in her assignment: "Since the probability of the union is the sum of the probabilities, we have that $P(L \cup D) = 0,60 + 0,55 = 1,15$; therefore, it is $100\,\%$ certain that they read in some way and there is $15\,\%$ left over". Identify and explain the two conceptual errors made by Lucía based on Kolmogorov's axioms. Next, knowing that $35\,\%$ of the students read in both formats ($P(L \cap D) = 0,35$), calculate the actual probability that a student reads in at least one of the two formats and the probability that they read in neither format ($P(\overline{L} \cap \overline{D})$).

Let $A$ and $B$ be two events associated with a random experiment. It is known 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 $P(B) = 0,4$.

Calculate, giving reasons, the following probabilities:

1. a) $P(A \cup B)$
2. b) $P(A \cap B)$
3. c) $P(\bar{A} | B)$
4. d) Determine whether events $A$ and $B$ are independent.

On a streaming platform, $60 \%$ of young subscribers watch sci-fi series (event $S$), $45 \%$ watch nature documentaries (event $D$) and $20 \%$ of users watch both types of content.

Calculate the probability that a user chosen at random:

1. a) Watches sci-fi series or nature documentaries.
2. b) Watches neither of the two genres.
3. c) Watches only one of the two genres.
4. d) Given that they do not watch sci-fi series, what is the probability that they watch nature documentaries?

Observe the following Venn diagram representing the probabilities of different regions for two events $M$ and $N$ in a sample space $E$.

From the data in the diagram, calculate the following probabilities:

1. a) $P(M \cup N)$
2. b) $P(M - N)$, that is, the probability of $M$ occurring but not $N$.
3. c) $P(\bar{M} \cup \bar{N})$ using de Morgan's Laws.
4. d) Are events $M$ and $N$ mutually exclusive? Justify your answer.

An eSports team is analysing its performance on two different maps in a video game: Inferno (event $I$) and Mirage (event $M$). Statistics show that they win on Inferno with a probability of $0,7$ and that the probability of winning on Mirage is $0,5$. It has been verified that the probability of winning on at least one of the two maps is $0,85$.

1. a) Calculate the probability that they win on both maps.
2. b) Is winning on Inferno independent of winning on Mirage?
3. c) Calculate the probability that they win on Mirage given that they lost on Inferno.

Given two events $A$ and $B$ from the same random experiment, it is known that:

• The probability of $A$ occurring is three times that of its complement, $\bar{A}$.

• $P(B) = 0,4$.

• $P(A \cap B) = 0,3$.

Calculate, giving reasons:

1. a) $P(A)$.
2. b) $P(A \cup B)$.
3. c) $P(\bar{A} \cap B)$.
4. d) $P(\bar{B} | \bar{A})$.

In a study on sustainable mobility among Sixth Form students, it was found that the probability of a student using a bicycle to get to school is $P(B) = 0,25$, that of using public transport is $P(T) = 0,60$ and that of using neither of these two modes of transport is $P(\bar{B} \cap \bar{T}) = 0,30$.

1. a) Are the events "using a bicycle" and "using public transport" compatible?
2. b) Calculate the probability that a student uses only a bicycle.
3. c) What is the probability that they use public transport given that they do not use a bicycle?

Consider two independent events $X$ and $Y$. It is known that $P(X) = 0,6$ and $P(X \cup Y) = 0,8$. Using the properties of probability and the definition of independence, calculate:

1. a) The probability of event $Y$.
2. b) The probability that $X$ occurs but $Y$ does not occur.
3. c) The probability that neither of the two occurs.

A task management app classifies notifications according to their urgency ($U$) and their type: personal ($P$) or work ($T$). It is known that $P(U) = 0,35$, $P(P) = 0,55$ and $P(U \cup P) = 0,75$.

Determine:

1. a) Whether events $U$ and $P$ are independent.
2. b) The probability that a notification is urgent given that it is personal.
3. c) The probability that a notification is personal but not urgent.
4. d) If notifications are either personal ($P$) or work ($T$), calculate the probability that a notification is not urgent and is for work, that is, $P(\bar{U} \cap T)$.

From a sample of workers in a Spanish metropolitan area, a mobility study analyzes the use of public transport ($T$) and private vehicles ($V$). The data indicate that $P(T) = 0,45$, $P(V) = 0,30$, and that the probability that a worker combines both means of transport in their daily commute is $P(T \cap V) = 0,10$.

1. a) Model the situation and calculate the probability that a randomly chosen person uses exclusively one of the two analyzed means of transport.

1. b) Express in formal language the event "uses neither of these two means of transport", calculate its probability using De Morgan's laws justifying each equality, and explain what this group represents in the context of the problem.

1. c) Reason whether the events $T$ and $V$ are mutually exclusive (incompatible) and explain what practical implication it would have for urban traffic if they were.

In a study on the digital entertainment market in Spain, two subscription services are analyzed: film and series platforms ($A$) and sports broadcasting platforms ($B$). It is known that $P(A) = 0,65$, that $P(B) = 0,40$, and that the probability of having neither of the two subscribed is $P(\overline{A} \cap \overline{B}) = 0,15$. You are asked to find the probability that a household has both services, $P(A \cap B)$.

To solve this problem, two approaches are proposed:

• Method 1: Apply De Morgan's law to solve for $P(A \cup B)$ from $P(\overline{A} \cap \overline{B})$ and substitute into the general addition formula for the union.

• Method 2: Construct a contingency table of probabilities ($2 \times 2$) identifying the four fundamental disjoint events until the required value is deduced.

1. a) Develop the complete solution applying the method you consider most direct and efficient, providing a reasoned justification for the mathematical steps followed.

1. b) Critically compare both methods and explain which one is more advantageous if, additionally, you were asked to calculate the probability of subscribing only to film and series ($P(A \cap \overline{B})$).

In a study on social media use among Year 13 students, two platforms are analysed: InstaPlus (event $A$) and TikToker (event $B$). It is known that the probability of a student using at least one of the two is $0,8$. Furthermore, the probability of using the first but not the second is $0,3$. If it is known that the probability of using both platforms is half the probability of using neither of them, calculate:

1. a) The probability that a student, chosen at random, uses the TikToker platform.

1. b) Are events $A$ and $B$ independent? Justify your answer mathematically by comparing $P(A \cap B)$ with the product of their individual probabilities.

1. c) Calculate the probability that they use only one of the two platforms.

Let $A$ and $B$ be two events of a random experiment such that their probabilities are given in terms of a parameter $k$. It is known that $P(A) = k$, $P(B) = 0,5$ and $P(A \cup B) = 0,7$.

1. a) Determine the value of $k$ for events $A$ and $B$ to be mutually exclusive.

1. b) Determine the value of $k$ for events $A$ and $B$ to be independent.

1. c) If $k = 0,4$, calculate the probability of $B$ occurring given that $A$ has not occurred, that is, $P(B | \bar{A})$.

A cybersecurity system monitors two types of attacks: Phishing ($A$) and Malware ($B$). The metrics for the last month indicate that $P(\bar{A} \cap \bar{B}) = 0,15$ and $P(A \cap B) = 0,25$. If the probability of suffering a Phishing attack is double that of suffering a Malware attack ($P(A) = 2P(B)$), find:

1. a) The probability of suffering a Malware attack.

1. b) The probability that, having suffered a Malware attack, it is not a Phishing attack.

1. c) Using de Morgan's Laws, demonstrate whether the union of the complements $P(\bar{A} \cup \bar{B})$ or the complement of the intersection $\overline{P(A \cap B)}$ is more likely.

In an eSports tournament, a player has a probability $p$ of winning the first match. If they win the first match, their confidence increases and the probability of winning the second match is $0,7$. If they lose the first match, the probability of winning the second match drops to $0,3$. Given that the probability of winning the second match is $0,54$, calculate:

1. a) The value of the probability $p$.

1. b) The probability that the player won the first match given that they lost the second match.

Formally prove, using Kolmogorov's axioms and the properties of probability, the following statement: "If $A$ and $B$ are two independent events, then their complements $\bar{A}$ and $\bar{B}$ are also independent".

Hint: You must start from the definition of independence $P(A \cap B) = P(A) \cdot P(B)$ and arrive at $P(\bar{A} \cap \bar{B}) = P(\bar{A}) \cdot P(\bar{B})$.

A music streaming app classifies its users into three categories according to their listening time: Free ($F$), Premium ($P$) and Family ($S$). $50\%$ are Free users, $30\%$ are Premium users and the rest are Family users. It is known that the percentage of users who regularly listen to podcasts is $20\%$ in the Free category, $60\%$ in the Premium category and $40\%$ in the Family category.

1. a) If we choose a user at random who listens to podcasts, what is the probability that they belong to the Premium plan?

1. b) If a new condition is added: "No Family user who listens to podcasts is under 18 years old". If we choose a user at random, what is the probability that they are a Family user, listen to podcasts and are under 18 years old? Justify your answer using the impossible event.

Let $A$, $B$ and $C$ be three events such that $A$ and $B$ are independent of each other, and $C$ is an event mutually exclusive with $A$ and with $B$. The following probabilities are known: $P(A) = 0,4$, $P(B) = 0,3$ and $P(C) = 0,1$. Calculate, giving reasons:

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

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

1. c) $P(\bar{A} | \bar{B})$

In a university entrance exam, $75\%$ of students pass Mathematics ($M$) and $60\%$ pass History ($H$). It is known that $10\%$ of students pass neither of the two subjects.

1. a) Calculate the probability that a student passes both subjects.

1. b) Given that a student has passed History, what is the probability that they have also passed Mathematics?

1. c) What is the probability that a student passes Mathematics but not History? Explain the relationship between this result and the difference of events $P(M - H)$.

A quality control department inspects garments in a textile factory prior to distribution. Two defects of independent origin are evaluated: fabric flaw ($A$) and dyeing defect ($B$). The plant records indicate that $P(A) = 0,70$ and that $P(B) = 0,55$.

1. a) Prove rigorously, relying on Kolmogorov's axioms, why it is impossible for events $A$ and $B$ to be mutually exclusive (incompatible) in this production.

1. b) Inverse problem: Reasonably determine the interval of admissible values $[k_{\min}, k_{\max}]$ for the probability of the intersection $k = P(A \cap B)$ so that the probability assignment remains mathematically consistent.

1. c) For each of the two endpoints of said interval ($k_{\min}$ and $k_{\max}$), calculate the probability that a garment turns out completely flawless ($P(\overline{A} \cap \overline{B})$) and interpret what happens on the garment manufacturing line in each of these extreme cases.

The National Statistics Institute (INE) studies the educational and employment status of the youth population in Spain using two events: $E$ ("the person is enrolled in formal education") and $T$ ("the person has paid employment"). The following data are available: the probability of being enrolled in education is $P(E) = 0,60$, the probability of having employment is $P(T) = 0,40$, and the proportion of young people who neither study nor work (event $\overline{E} \cap \overline{T}$) is $0,15$.

1. a) Prove using the properties of probability that $P(E \cup T) = P(E) + P(T \cap \overline{E})$, by decomposing the union into disjoint events. From this relation and the problem's data, deduce the exact value of $P(E \cap T)$ and that of $P(T \cap \overline{E})$.

1. b) A ministerial youth report projects a scenario in which the student rate remains fixed at $P(E) = 0,60$, but the rate of young people with no occupation is reduced to $P(\overline{E} \cap \overline{T}) = 0,08$. If in that new scenario it were required that no student combine studies with work ($E$ and $T$ mutually exclusive), reason using the axioms of probability whether such a situation would be possible or if the proposal contains a mathematical contradiction.