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.

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 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.

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.

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.

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.

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.

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 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?