Basic concepts

An $8$-sided die in the shape of a regular octahedron, numbered from $1$ to $8$, is rolled. Write down the sample space $E$ associated with this random experiment and determine the outcomes that make up the following events:

• $A =$ "Obtaining a prime number".

• $B =$ "Obtaining a number greater than $5$".

• $C = A \cap B$.

$3$ fair coins are tossed simultaneously. Write down the complete sample space $E$ using the notation $C$ for heads and $X$ for tails. Next, calculate the probability of getting exactly two heads.

In a technology college with $150$ students, a survey was carried out on the use of programming languages. The results show that $85$ students program in Python ($P$), $60$ program in Java ($J$) and $25$ program in both languages.

If a student is selected at random from this college, find the following probabilities:

1. a) That they program in at least one of the two languages.
2. b) That they do not program in either of these two languages.
3. c) That they program in Python, but not in Java.

In a veterinary clinic, $55\%$ of the dogs treated are vaccinated against rabies ($R$), $40\%$ are vaccinated against parvovirus ($P$) and $25\%$ are vaccinated against both diseases. A dog's record is chosen at random. Calculate the probability that it:

1. a) Is vaccinated against rabies but not against parvovirus, that is, $P(R - P)$.
2. b) Is vaccinated against exactly one of the two diseases.

In a meteorological study, the probability that it rains on Saturday is $P(S)$ and that it rains on Sunday is $P(D)$. Knowing that the probability that it rains on at least one of the two days of the weekend is $P(S \cup D) = 0.85$, use De Morgan's laws or the properties of the complementary event to calculate the probability that it does not rain on either day of the weekend, that is, $P(\overline{S} \cap \overline{D})$.

In a meteorological study, the following events are recorded for a specific autumn day: $L =$ "it rains" and $V =$ "there are strong wind gusts", with known probabilities $P(L) = 0.40$, $P(V) = 0.30$, and $P(L \cap V) = 0.15$.

Aoife needs to calculate the probability that on a given day "it does not rain and there is no strong wind either". In her notebook, she writes:

"The opposite event of neither of the two things happening is that both happen at the same time, that is, the complement of the intersection: $\overline{L \cap V}$. Therefore: $P(\overline{L \cap V}) = 1 - P(L \cap V) = 1 - 0.15 = 0.85$".

1. a) Identify the conceptual error made by Aoife when translating the verbal statement into set operations and explain with justification why her logical equivalence is false.
2. b) Write the correct set-theoretic expression for the event "it does not rain and there is no wind" and use one of De Morgan's laws to find its correct probability.

In a microchip manufacturing plant in Ireland, each component undergoes three automated quality control checks. The failure events are defined as: $F_1 =$ "fails the electrical test", $F_2 =$ "fails the thermal test", and $F_3 =$ "fails the optical test". Historical control data are: $P(F_1) = 0.08$; $P(F_2) = 0.06$; $P(F_3) = 0.05$; $P(F_1 \cap F_2) = 0.02$; $P(F_1 \cap F_3) = 0.015$; $P(F_2 \cap F_3) = 0.01$, and $P(F_1 \cap F_2 \cap F_3) = 0.003$.

The quality engineering team needs to determine the probability that a component successfully passes at least one of the three tests (that is, that it does not fail all of them simultaneously).

To calculate this probability, two technicians propose different approaches:

• Cian's method: Expresses the event as $\overline{F_1} \cup \overline{F_2} \cup \overline{F_3}$ and proposes directly applying the formula for the union of three events: $P(\overline{F_1} \cup \overline{F_2} \cup \overline{F_3}) = P(\overline{F_1}) + P(\overline{F_2}) + P(\overline{F_3}) - P(\overline{F_1} \cap \overline{F_2}) - P(\overline{F_1} \cap \overline{F_3}) - P(\overline{F_2} \cap \overline{F_3}) + P(\overline{F_1} \cap \overline{F_2} \cap \overline{F_3})$.

• Aoife's method: Applies De Morgan's laws generalized to three events to transform the union of complements into the complement of a direct intersection, calculating the result in a single step.

1. a) Solve the problem using Aoife's method and calculate the final numerical value.
2. b) Without needing to compute all the numbers in Cian's method, explain what operational difficulty it presents: how would Cian have to calculate probabilities such as $P(\overline{F_1} \cap \overline{F_2})$ or $P(\overline{F_1} \cap \overline{F_2} \cap \overline{F_3})$ from the given data?
3. c) Compare both methods: justify which one is more efficient and explain why switching to the complement is a fundamental technique for simplifying probability problems.

The random experiment of rolling a fair cubic die numbered from $1$ to $6$ is performed. Consider the events $A =$ "Obtaining an even number" and $B =$ "Obtaining a prime number greater than $2$".

Write down the outcomes of each event and determine, justifying your answer, whether events $A$ and $B$ are compatible or incompatible.

In an environmental toxicology study in a river, three pollutants are monitored. Let $A$, $B$ and $C$ be the events of finding dangerous levels of the respective pollutant. It is known that $P(A) = \frac{1}{2}$, $P(B) = \frac{1}{3}$ and $P(C) = \frac{1}{4}$.

Furthermore, it is known that pollutants $A$ and $B$ never appear together in dangerous levels (they are incompatible events), and that whenever pollutant $C$ is detected, pollutant $B$ is also detected (that is, $C$ is a subset of $B$, $C \subset B$).

Calculate the probability of finding neither pollutant $A$ nor pollutant $B$, that is, $P(\overline{A} \cap \overline{B})$, and the probability of finding at least one of the three pollutants, $P(A \cup B \cup C)$.

The symmetric difference of two events $A$ and $B$, denoted as $A \triangle B$, is defined as the event that $A$ or $B$ occurs, but not both simultaneously. Algebraically, $A \triangle B = (A \cap \overline{B}) \cup (\overline{A} \cap B)$.

Prove the relation $P(A \triangle B) = P(A \cup B) - P(A \cap B)$. Next, knowing that $P(A \cup B) = 0.85$, $P(\overline{A}) = 0.3$ and $P(B) = 0.6$, calculate the numerical value of $P(A \triangle B)$.

Two fair cubic dice are rolled, one red and one blue. The score on the top face of each is recorded. Calculate the probability of the event $A =$ "The sum of the scores is exactly $8$" or the event $B =$ "The absolute value of the difference between the scores is exactly $2$" occurring. Are $A$ and $B$ incompatible events?

On a digital entertainment platform with $500$ subscribers, preferences for three genres are analysed: Action ($A$), Comedy ($C$) and Drama ($D$). The data reveal that:

• $220$ watch Action, $180$ watch Comedy, $150$ watch Drama.

• $70$ watch Action and Comedy, $50$ watch Action and Drama, $40$ watch Comedy and Drama.

• $15$ watch all three genres.

If a subscriber is chosen at random, find the probability that they:

1. a) Watch Action movies exclusively.
2. b) Watch exactly one of the three genres.
3. c) Watch none of these three genres.

Based on youth urban mobility data published by the National Transport Authority in Ireland, how students in a municipality commute is analyzed. The following events are considered:

$M =$ "regularly uses the metro",

$A =$ "regularly uses the city bus",

$B =$ "uses a personal bicycle or electric scooter".

The records show the following probabilities: $P(M) = 0.45$; $P(A) = 0.50$; $P(B) = 0.30$; $P(M \cap A) = 0.20$; $P(M \cap B) = 0.12$; $P(A \cap B) = 0.15$, and the probability of combining all three modes of transport is $P(M \cap A \cap B) = 0.05$.

1. a) Model the situation by expressing the following events in set notation (using unions, intersections, or complements): $S_1 =$ "uses at least one of the three modes of transport", $S_2 =$ "uses none of these three modes of transport", and $S_3 =$ "uses exclusively the metro".
2. b) Calculate the probability of $S_1$ by applying the general formula for the union of three events.
3. c) Find the probability of $S_2$ and of $S_3$, detailing the intermediate steps.

A risk analysis laboratory evaluates three critical scenarios $A$, $B$, and $C$ in an infrastructure. The following information regarding their probabilities is available:

• Events $A$ and $B$ are mutually exclusive ($A \cap B = \emptyset$).

• Events $B$ and $C$ are mutually exclusive ($B \cap C = \emptyset$).

• Events $A$ and $C$ are compatible, with $P(A \cap C) = 0.10$.

• The individual probabilities depend on a parameter $p \in \mathbb{R}$: $P(A) = 2p$, $P(B) = 3p$, and $P(C) = 0.35$.

1. a) Algebraically show how the general formula for the union of three events, $P(A \cup B \cup C)$, simplifies when substituting the given conditions of mutual exclusivity.
2. b) It is known that the probability that none of the three risk scenarios occurs is exactly $0.15$. Set up an equation using De Morgan's laws and find the value of parameter $p$.
3. c) With the value of $p$ obtained, analyze whether it is mathematically viable for a fourth risk scenario $D \subseteq E$ to exist that is completely mutually exclusive with $A \cup B \cup C$ and whose probability is $P(D) = 0.20$. If it is not possible, deduce with justification what the strict upper bound for $P(D)$ is and what set inclusion relationship exists between $D$ and the complement $\overline{A \cup B \cup C}$.

Let $A$ and $B$ be two events in the same random experiment. The following probabilities are known: $P(A) = 0.5$, $P(B) = 0.4$ and the probability that both occur simultaneously is $P(A \cap B) = 0.15$.

Calculate the probability that at least one of the two events occurs, that is, $P(A \cup B)$, and the probability that $A$ does not occur, denoted as $P(\overline{A})$.

On an electric motor assembly line, it is known that the probability that a motor has a mechanical defect is $P(M) = 0.06$, the probability that it has an electrical defect is $P(E) = 0.04$, and the probability that it has both defects simultaneously is $P(M \cap E) = 0.01$. Calculate the probability that a randomly chosen motor:

1. a) Has at least one defect.

1. b) Has no defects at all.

1. c) Has a mechanical defect given that it has no electrical defect, that is, $P(M \mid \overline{E})$ (you can use the fact that $P(M \cap \overline{E}) = P(M) - P(M \cap E)$).

A biased cubic die has been manufactured such that the probability of each face landing face up is directly proportional to the number on that face. That is, $P(x) = k \times x$ for $x \in \{1, 2, 3, 4, 5, 6\}$, where $k$ is a real constant.

1. a) Calculate the exact value of the constant $k$ knowing that the sum of the probabilities of the sample space must be $1$.
2. b) Calculate the probability of obtaining a prime number when rolling this biased die.

A computer engineering student is configuring a new server. They can choose between $3$ processor models, $4$ different RAM capacities and $2$ types of hard drive (SSD or NVMe). Applying the multiplication principle, how many different server configurations can they create by choosing exactly one component of each type? If they choose a configuration at random, what is the probability that it has an NVMe drive?

To access a secure database, an employee must generate a $4$-digit numeric PIN code (from $0$ to $9$). The system allows digits to be repeated. Calculate the probability that a completely randomly generated PIN satisfies the following condition: the first digit is a $7$ and the last digit is even (considering $0$ as even).

An opaque urn contains $4$ white balls and $6$ black balls. $2$ balls are to be drawn consecutively.

1. a) Calculate the probability that both balls are white if the drawing is done with replacement (the first ball is returned to the urn before drawing the second).
2. b) Calculate the probability that both balls are white if the drawing is done without replacement.