Definitions

Let $A$ and $B$ be two events in the same random experiment such that $P(A) = 0.4$, $P(B) = 0.3$ and $P(A \cup B) = 0.7$. Justify mathematically whether events $A$ and $B$ are compatible or incompatible based on the axioms of probability.

In a Mathematics test, a random experiment is proposed where two events have probabilities $P(A) = 0.65$ and $P(B) = 0.55$. Seán writes the following solution on his exam:

"Applying Kolmogorov's third axiom, the probability that at least one of the two occurs is $P(A \cup B) = P(A) + P(B) = 0.65 + 0.55 = 1.20$. Since this result is greater than $1$ and contradicts the sure event axiom, I deduce that the data in the problem are erroneous and that such events cannot coexist in any real experiment."

1. a) Identify and explain with reasoning the two conceptual errors that Seán makes regarding the conditions required by the third axiom and the definition of compatible events.
2. b) Without modifying the values of $P(A)$ and $P(B)$, correct Seán's solution by determining the interval of possible values for $P(A \cap B)$ that guarantee that the experiment is mathematically consistent.

A game consists of tossing a fair coin repeatedly until the first "Heads" ($C$) appears, with a maximum limit of 3 tosses (if "Tails" ($X$) appears on the third toss, the game ends anyway).

• Write down the sample space $E$ of this experiment.

• Assign the corresponding probability to each elementary event, justifying your answer.

A fair 12-sided (dodecahedral) die with its faces numbered from 1 to 12 is rolled. The outcome on the top face is recorded. List the elements of the sample space $E$ and the following events:

• $A$: "Obtaining a multiple of 3".

• $B$: "Obtaining a prime number".

• $C$: "Obtaining a number greater than 10".

An opaque urn contains 5 red balls, 3 blue balls and 2 green balls, all identical to the touch. If a ball is drawn at random, calculate the probability of:

• Drawing a blue ball.

• Not drawing a red ball.

• Drawing a ball that is red or green.

A card is drawn at random from a 40-card deck. Using Laplace's rule, calculate the probability of the following elementary and compound events:

• $A$: "Drawing a Knight".

• $B$: "Drawing a Cup card".

• $C$: "Drawing a Gold picture card" (the picture cards are Jack, Knight and King).

Two fair cubic dice are rolled simultaneously, one red and one blue. Using the two-dimensional sample space, calculate the probability that:

• The sum of the scores is exactly 8.

• The absolute value of the difference between the scores is 2.

An urn contains $n$ white balls and 6 black balls. A ball is drawn at random. Knowing that the probability of drawing a white ball is $\frac{3}{5}$, set up an equation to find the value of $n$ and determine the total number of balls in the urn.

In a cybersecurity system, a user must create a 4-digit numeric PIN (from 0 to 9). If the system generates a random PIN, calculate the probability that the PIN contains exactly two equal digits (for example, 1213 or 9904, but not 2225 nor 1122). Use Laplace's rule and combinatorial techniques.

In a random experiment, the sample space is $E = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$. Given the events $A = \{2, 4, 6, 8, 10\}$ and $B = \{1, 2, 3, 4, 5\}$, list the elements of the following events resulting from operations on them:

• $A \cup B$

• $A \cap B$

• $A - B$

• $\overline{A} \cap B$

In an electronic components factory, it is known that the probability of a chip having a soldering defect is $0.05$, the probability of having a packaging defect is $0.03$, and the probability of having both defects simultaneously is $0.01$. Calculate the probability that a randomly chosen chip has at least one of the two defects.

At a meteorological centre, the event $L$: "It rains tomorrow" and the event $V$: "It is windy tomorrow" are defined. Express the meaning of the following algebraic events in ordinary language:

• $\overline{L} \cap \overline{V}$

• $L \cup V$

• $V - L$

In a clinical study on a new viral strain, a group of patients is observed. The events $F$: "The patient has a fever" and $T$: "The patient has a cough" are defined. Express the following events described in natural language in set notation (intersections, unions, complements):

• The patient has at least one of the two symptoms.

• The patient has a fever but no cough.

• The patient has neither of the two symptoms.

• The patient has exactly one symptom.

In a software development company, the probability that a program has a user interface error (event $U$) is $0.45$. The probability that it has a database error (event $D$) is $0.30$. If the probability that it has a user interface error but not a database error is $0.25$, calculate the probability that the program has no errors.

A financial analyst evaluates two investment funds, $A$ and $B$. They determine that the probability that fund $A$ is profitable is double the probability that fund $B$ is profitable. Furthermore, they know that the probability that at least one is profitable is $0.85$ and the probability that both are profitable is $0.20$. Set up a system of equations and calculate the probability that fund $A$ is profitable.

A random experiment is carried out consisting of rolling two fair cubic dice (with faces numbered from 1 to 6) and the event $S$ is defined: "obtain at least one number 5 between the two dice". To calculate $P(S)$, two students propose different methods based on the axioms:

• Method 1 (Via complementary event): Calculate the probability of the complementary event $\overline{S}$ ("obtain no 5 on either die") and apply $P(S) = 1 - P(\overline{S})$.

• Method 2 (Via general union): Define $A$: "a 5 appears on the first die" and $B$: "a 5 appears on the second die", and calculate $P(S) = P(A \cup B) = P(A) + P(B) - P(A \cap B)$.

1. a) Fully develop both methods, justifying at each step which axiom or probability property supports each equality and applying Laplace's rule. Verify that both procedures lead to the same result.
2. b) Imagine that the game is generalized to the simultaneous roll of 6 fair dice and the probability of obtaining at least one 5 is sought. Analyze and justify which of the two methods is clearly more efficient and scalable, explaining the algebraic difficulties that applying the other would entail.

In logic circuit design, the activation of a relay depends on events $A$ and $B$ according to the Boolean expression that defines event $K$: $K = (A \cup B) \cap (\overline{A} \cup B) \cap (A \cup \overline{B})$.

Analytically simplify the expression for event $K$ using the properties of the algebra of events. Afterwards, calculate $P(K)$ knowing that $P(A) = 0.6$, $P(B) = 0.5$ and $P(A \cup B) = 0.8$.

From Met Éireann's historical series in a mountain area, three events are defined for a winter day: $L$ ("precipitation in the form of rain or snow"), $N$ ("sky with compact cloudiness greater than $80\,\%$") and $V$ ("strong wind gusts exceeding $70\text{ km/h}$"). The data support that $P(L) = 0.65$, $P(N) = 0.80$, and $P(V) = 0.35$. Furthermore, physical laws imply that precipitation cannot occur without such compact cloudiness ($L \subseteq N$).

1. a) Formally prove, using only Kolmogorov's axioms and the decomposition of $N$ as the union of disjoint events, that $P(L \cap N) = P(L)$ and that $P(\overline{L} \cap N) = P(N) - P(L)$. Calculate the numerical value of both probabilities.
2. b) Without assuming any prior physical dependence or independence relationship between the wind $V$ and the cloudiness $N$, deduce from the axioms the narrowest possible numerical interval $[m, M]$ within which the probability $P(N \cap V)$ must certainly lie.
3. c) A preliminary meteorological report claims that "the probability of rain or strong wind is $P(L \cup V) = 0.20$". Prove by logical contradiction, relying on the monotonicity property or on the intersection bounds, why this statement is mathematically impossible.

Look at the following Venn diagram representing a sample space $E$ and two events $A$ and $B$. Knowing that the area of each region is proportional to its probability, express the region shaded in green in set notation and calculate its probability if $P(A) = 0.55$, $P(B) = 0.40$ and $P(A \cap B) = 0.15$.

In a school, $60\%$ of 1st Baccalaureate students take Physics, $50\%$ take Technical Drawing and $20\%$ take neither of the two. If a student is chosen at random, determine the following probabilities using axiomatic properties and De Morgan's Laws:

• That they take both subjects.

• That they take Physics but not Technical Drawing.

• That they take exactly one of the two subjects.

According to the Survey on Equipment and Use of Information and Communication Technologies in Households published by the CSO, in a given county the probability that a randomly selected household has a fiber-optic internet connection ($A$) is $P(A) = 0.82$, and the probability that it has a subscription to a video-on-demand platform ($B$) is $P(B) = 0.64$. Furthermore, the study indicates that the probability that a household does not have either of these two services is $0.06$.

1. a) Model the given information by expressing each piece of data in the formal language of set theory and probability (identifying complements, unions, and intersections).
2. b) Prove step by step, using exclusively Kolmogorov's axioms and the properties derived from them, whether events $A$ and $B$ are compatible or incompatible.
3. c) Formally deduce the relationship between $P(A - B)$ and the known probabilities, and calculate the probability that a randomly chosen household has fiber optics but lacks a video platform.

In a secondary school, the administration conducts a survey on the sustainable mobility of its Leaving Certificate students. The results indicate that the probability that a randomly chosen student uses the metropolitan bus pass ($B$) is $P(B) = 0.75$, and the probability that they use the municipal bike lane ($C$) is $P(C) = 0.40$.

1. a) Before posing any algebraic calculation, estimate intuitively: is it possible that no student uses both means of transport at the same time? What do you intuit must be the minimum percentage of students who necessarily combine both means?
2. b) Apply the general formula for the probability of the union, $P(B \cup C) = P(B) + P(C) - P(B \cap C)$, along with the axiom that no probability can exceed $P(E) = 1$, to calculate the minimum bound of $P(B \cap C)$ and also determine its maximum bound. Compare these results with your previous estimate and assess whether your intuition was correct.

In a quality control check, two tests are applied to a batch of products. It is known that the probability of passing test $A$ is $P(A) = 0.85$ and the probability of passing test $B$ is $P(B) = 0.75$. Without knowing the probability of the intersection, determine algebraically the maximum value and the minimum value that the probability of a product passing both tests, $P(A \cap B)$, can take.

A quality control team analyzes two operational stoppage events, $A$ and $B$, on an assembly line. The estimated probabilities depend on a design parameter $k \in \mathbb{R}$, such that $P(A) = k$, $P(B) = 2k - 0.1$, and it is known that the probability that the line stops due to at least one of the two causes is $P(A \cup B) = 0.70$. Due to monitoring requirements, both events are required to be able to coexist with a probability verifying $P(A \cap B) \geq 0.10$.

1. a) Using Kolmogorov's axioms, the addition property for compatible events, and the universal bounds that $P(A \cap B)$ must satisfy with respect to $P(A)$ and $P(B)$, deduce with reasoning the interval of admissible values for the parameter $k$.
2. b) For the minimum admissible value of $k$ obtained in the previous section, pose an inverse design problem: construct a discrete and finite sample space $E$ with the smallest possible number of equiprobable elementary events (to apply Laplace's rule) and define by extension the subsets representing $A$, $B$, and their intersection, verifying that they reproduce the calculated probabilities exactly.