Definitions of Probability

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.

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".

A card is drawn at random from a 40-card Spanish 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).

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.

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 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.

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.

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:

• $A \cup B$

• $A \cap B$

• $A - B$

• $\overline{A} \cap B$

At a meteorological center, 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 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 $P(A \cup B) = 0.8$.

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.

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.

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.

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 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.