Exercises: Events

A high school senior is organizing their digital note folders. In their "Economics" folder, they have 12 PDF files, 8 text documents, and 5 diagrams in image format. If they choose a file at random to review before an exam, determine:

1. a) The sample space of the experiment.
2. b) The probability that the chosen file is a diagram in image format.
3. c) The probability that the chosen file is not a text document.

In a high school senior class, a survey on social media use has been carried out. It is known that event $A$ is "using Instagram" and event $B$ is "using TikTok". Define the following situations using event operations (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.

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) $P(A \cup B)$
2. b) $P(\bar{A})$ (probability of the complementary event of $A$).
3. c) $P(\bar{A} \cap \bar{B})$ using De Morgan's Laws.

An esports team has players of three nationalities on its roster: 5 Spaniards, 3 Frenchmen and 2 Italians. If a captain is selected at random 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.

Given two events $M$ and $N$ such that they are incompatible with each other, with $P(M) = 0,35$ and $P(N) = 0,45$, find the probability that:

1. a) Event $M$ or event $N$ occurs.
2. b) Neither of the two events occurs.
3. c) $M$ and $N$ occur simultaneously.

On a streaming platform, $70\%$ of users watch action series ($S$), $40\%$ watch documentaries ($D$) and $25\%$ watch both types of content. Calculate the probability that a user chosen at random:

1. a) Watches action series or documentaries.
2. b) Watches neither of these two contents.

Let $A$ and $B$ be two events such that $P(A) = 0,5$, $P(\bar{B}) = 0,7$ and $P(A \cup B) = 0,65$. Calculate the probability of the intersection $P(A \cap B)$ and determine whether the events are compatible.

We draw a ball from an urn containing 10 balls numbered from 1 to 10. We consider the events:

$A = \{ \text{drawing an even number} \}$

$B = \{ \text{drawing a multiple of 3} \}$

1. a) Write 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 $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.

Let $A$ and $B$ be two events of 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) Indicate with reasoning whether events $A$ and $B$ are compatible or incompatible.

In a study on the consumption of digital content among 16-year-olds, it has been detected that $70 \%$ use the platform StreamVibe and $45 \%$ use CloudMusic. Furthermore, it is known that $25 \%$ of the young people do not use either of the two platforms. If we choose a young person at random, calculate:

1. a) The probability that they use both platforms.
2. b) The probability that they use only StreamVibe.
3. c) The probability that they use CloudMusic but not StreamVibe.

We have the following partial contingency table showing the distribution of a group of 120 high school students according to their participation in extracurricular activities (Sports or Music):

1. a) Complete the contingency table in your notebook.
2. b) Calculate the probability that a student chosen at random does not do either of the two activities.
3. c) Are the events "Participating in Music" and "Participating in Sports" independent? Justify your answer mathematically.

A high school senior 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 do not know either.
3. c) The probability that they know at least one of the two selected topics.

Let $A$ and $B$ be two events such that $P(A) = 0,4$ and $P(B) = 0,3$. Calculate the probability of the union $P(A \cup B)$ in each of the following scenarios:

1. a) $A$ and $B$ are incompatible events.
2. b) $A$ and $B$ are independent events.
3. c) Event $B$ is contained in event $A$ ($B \subset A$).

In a role-playing game, a player must pass two consecutive tests. The probability of passing the first test is $0,7$. If they pass the first, the probability of passing the second is $0,8$, but if they fail the first, the probability of passing the second drops to $0,4$.

1. a) Draw the tree diagram that represents this situation.
2. b) Calculate the probability that the player passes both tests.
3. c) Calculate the probability that the player passes only one of the two tests.

From a 40-card Spanish deck, two cards are drawn successively without replacement. Determine the following probabilities:

1. a) That both cards are face cards (Jack, Knight, or King).
2. b) That the first card is an Ace and the second is a King.
3. c) That at least one of the two cards is a Coin.

Consider the events $A$ and $B$ such that $P(A) = 0,5$, $P(B|A) = 0,4$ and $P(A \cup B) = 0,65$. From this data:

1. a) Find the value of the probability of the intersection $P(A \cap B)$.
2. b) Calculate the value of $P(B)$.
3. c) Determine with reasoning whether the events $A$ and $B$ are independent.

On a streaming platform, it has been observed that $65\%$ of users under 20 years old watch animated series (event $A$) and $40\%$ watch nature documentaries (event $D$). Furthermore, it is known that the probability that a user does not watch animated series given that they do not watch nature documentaries is $0,5$.

Calculate with reasoning:

1. a) The probability that a user watches both types of content.
2. b) The probability that a user watches animated series but not nature documentaries.
3. c) Are events $A$ and $D$ independent? Justify your answer using the previous results.

A team of video game developers is analyzing the errors (bugs) of their latest version. They have classified the errors into three categories: Critical ($C$), Graphical ($G$) and Sound ($S$). After a massive test, the following data has been obtained:

• $10\%$ of the errors are of all three types at the same time.

• The probability that an error is Critical is $P(C) = 0,4$.

• The probability that it is Graphical is $P(G) = 0,5$.

• The probability that it is Sound is $P(S) = 0,35$.

• The probability that an error is Critical and Graphical is $0,2$.

• The probability that it is Graphical and Sound is $0,15$.

• The probability that an error does not belong to any of these three categories is $0,2$.

Calculate the probability that an error selected at random is:

1. a) Critical and Sound.
2. b) Exclusively a Graphical error.
3. c) Sound, given that it is neither Critical nor Graphical.

Let $A$ and $B$ be two events of 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?

In an esports championship, the plays of a pro player are analyzed. The events $K$ (getting a kill) and $W$ (winning the round) are defined. The analyst determines that:

• The probability of winning the round having gotten a kill is $P(W|K) = 0,8$.

• The probability of winning the round without having gotten any kills is $P(W|\bar{K}) = 0,3$.

• The probability that in a round the player wins or gets a kill is $0,85$.

Using the algebra of events, calculate:

1. a) The probability that the player gets a kill in a round, $P(K)$.
2. b) The probability that the player wins the round and does not get any kills.
3. c) Show whether the events "winning the round" and "getting a kill" are dependent or independent.

Prove, using the properties of probability and De Morgan's laws, the truth or falsity of the following statement:

"If $A$ and $B$ are two incompatible events with non-zero probabilities, then it is always true that $P(\bar{A} | B) = 1$".

Accompany your reasoning with an original numerical example where $P(A) = 0,3$ and $P(B) = 0,4$.

A cybersecurity system has two independent protection barriers, $B_1$ and $B_2$. The probability that an external attack overcomes the first barrier is $P(B_1) = 0,15$ and the probability that it overcomes the second is $P(B_2) = 0,10$.

Calculate the probability that an attack:

1. a) Overcomes at least one of the two barriers.
2. b) Overcomes exactly one of the barriers.
3. c) Is not detected by either barrier, given that the attack has managed to overcome the first protection.

In a study on the use of social networks, two high school seniors are selected consecutively and without replacement. It is known that $60\%$ of the students at the school use social network X.

If the probability that the second selected student uses social network X is $35/59$, determine the total number of high school seniors there are in that school.

Consider three events $A$, $B$ and $C$ such that $A$ and $B$ are independent, and $A$ and $C$ are incompatible. The following probabilities are known:

$P(A) = 0,4$; $P(B) = 0,5$; $P(C) = 0,2$ and $P(B \cap C) = 0,1$.

Find the probability of the following compound events:

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