Events and Probability Rules

In the study of everyday phenomena, it is essential to distinguish between certainty and chance. Classify the following experiments as random or deterministic:

1. a) Taking a penalty kick in training and observing if the ball goes into the goal.
2. b) Measuring the time it takes for an object to fall from a height of 5 feet in a vacuum.
3. c) Observing the number of WhatsApp messages you receive between 5:00 p.m. and 6:00 p.m. on any given Monday.
4. d) Calculating the total cost of buying 3 pounds of oranges if the price per pound is $1,25 \text{ \$}$.
5. e) Drawing a card from a standard deck and observing if it is clubs.

We have a wheel of fortune divided into 8 equal sectors, numbered from 1 to 8. The wheel is spun and the number indicated by the arrow is noted.

Based on this experiment:

1. a) Write the sample space $E$.
2. b) Indicate the elements that make up the event $A = \text{"getting a number greater than 5"}$.
3. c) Indicate the elements that make up the event $B = \text{"getting an even number"}$.
4. d) Describe in words the complementary event $\bar{A}$.

Two cubic dice (six-sided) are rolled and the sum of the points obtained is noted.

1. a) What is the sample space $E$ of this experiment?
2. b) Write the elementary events that make up the event $S = \text{"the sum is a multiple of 5"}$.
3. c) Determine whether the event $I = \text{"the sum is a number less than 2"}$ is a possible, impossible or certain event.

A group of friends analyzes their preferences regarding social media. Let $I$ be the event "using Instagram" and $T$ the event "using TikTok". Express the following situations symbolically using operations on events:

1. a) The friend uses Instagram or TikTok (or both).
2. b) The friend uses both social media platforms.
3. c) The friend uses Instagram but does not use TikTok.
4. d) The friend uses neither of the two social media platforms.

In a bag there are 10 balls numbered from 1 to 10. We consider the following events:

$A = \{1, 3, 5, 7, 9\}$

$B = \{6, 7, 8, 9, 10\}$

Find the elements of the following resulting events:

1. a) $A \cup B$
2. b) $A \cap B$
3. c) $\bar{A}$ (complementary event of $A$)
4. d) $A - B$ (events that are in $A$ but not in $B$)

Consider the experiment of drawing a card from a 40-card Spanish deck. Indicate whether the following pairs of events are compatible or incompatible:

1. a) $A = \text{"Drawing a King"}$ and $B = \text{"Drawing a Jack"}$.
2. b) $C = \text{"Drawing a Gold"}$ and $D = \text{"Drawing a Face Card"}$.
3. c) $E = \text{"Drawing an even number"}$ and $F = \text{"Drawing the Ace of Swords"}$.
4. d) $G = \text{"Drawing a Knight"}$ and $H = \text{"Drawing a card of Cups"}$.

Observe the following Venn diagram representing the events $A$ and $B$ within a sample space $E$.

If the point $x$ represents an outcome of the experiment, indicate whether the following statements are true or false based on the graph:

1. a) The outcome $x$ belongs to the event $A \cup B$.
2. b) The outcome $x$ belongs to the event $A \cap B$.
3. c) The events $A$ and $B$ are incompatible.
4. d) The outcome $x$ belongs to the event $A - B$.

In a high school classroom there are 12 students who have chosen French as a second language, 15 who have chosen German and 3 who have chosen both. If we choose a student at random:

1. a) What is the total number of students who study either of the two languages (union event)?
2. b) How many students study only German?
3. c) Using Laplace's Rule, if there are 30 students in the class in total, calculate the probability that the chosen student studies German.

In a survey carried out among 11th grade students about their subscriptions to streaming platforms, it was found that $60\%$ are subscribed to Netflix ($N$), $45\%$ to HBO Max ($H$) and $20\%$ are not subscribed to either of the two. If we choose a student at random, calculate:

1. a) The probability that they are subscribed to both platforms.
2. b) The probability that they are subscribed to Netflix but not to HBO Max ($P(N - H)$).
3. c) Represent the situation using a Venn diagram indicating the probabilities of each region.

Prove with reasoning, using De Morgan's Laws and the axiomatic properties of probability, the following equality for any two events $A$ and $B$:

$$P(\bar{A} \cap \bar{B}) - P(\bar{A}) = P(B) - P(A \cup B)$$

Then, interpret what this equality means in terms of the relationship between the events $A$ and $B$ when the result is equal to zero.

Observe the following Venn diagram representing the probabilities of the events $A$ (studying French) and $B$ (studying German) in a language center.

From the data in the graph, calculate:

1. a) $P(A \cup B)$
2. b) $P(\bar{A} | B)$
3. c) $P(\bar{A} \cap \bar{B})$ using De Morgan's Laws.

In a prestigious music academy, $60\%$ of the students play the piano ($P$), $45\%$ play the violin ($V$) and $20\%$ play neither of these two instruments. A student is selected at random.

1. a) Determine the probability that they play both instruments.
2. b) If we know that the selected student plays the violin, what is the probability that they do not play the piano?
3. c) Three students are chosen at random independently. Calculate the probability that at least one of them plays the piano but not the violin.

In a study on digital media consumption among teenagers, two events are analyzed: $A$ (the user regularly uses a microblogging platform) and $B$ (the user consumes content on video streaming platforms). After carrying out a survey, the following probability data have been obtained:

• The probability that a user does not use either of the two platforms is $P(\bar{A} \cap \bar{B}) = 0,15$.

• The probability that they regularly use the microblogging platform is $P(A) = 0,6$.

• The probability that they do not use the streaming platform given that they do use the microblogging platform is $P(\bar{B}/A) = 0,4$.

Calculate the following probabilities, showing your reasoning:

1. a) The probability that the user consumes streaming content, $P(B)$.
2. b) The probability that they use the microblogging platform but not the streaming platform, $P(A - B)$.
3. c) Are the events $A$ and $B$ independent? Justify your answer using calculations.

Given two events $A$ and $B$ of a random experiment, we know that the probability that neither of them occurs is $P(\bar{A} \cap \bar{B}) = 0,3$. If it is also known that $P(A) = 0,4$ and $P(B) = 0,5$, determine:

1. a) Are $A$ and $B$ mutually exclusive events? Justify your answer.
2. b) Are $A$ and $B$ independent events?
3. c) Calculate $P(A \cap \bar{B})$.

In an e-sports tournament, $70\%$ of the participants use a mechanical keyboard ($T$) and $40\%$ use an ergonomic mouse ($R$). We know that the probability that a participant chosen at random uses neither a mechanical keyboard nor an ergonomic mouse is $0,15$.

1. a) Calculate the probability that a player uses both.
2. b) If we know that a player uses a mechanical keyboard, what is the probability that they also use an ergonomic mouse?
3. c) Is the use of an ergonomic mouse independent of the use of a mechanical keyboard?

A data analyst studies the probability of an Instagram post being successful. They define the events $V$ (the post is a video) and $L$ (the post receives more than 500 "likes"). It is known that $P(V) = 0,25$, $P(L) = 0,40$ and $P(V \cup L) = 0,55$.

1. a) Calculate $P(V \cap L)$.
2. b) Determine whether the events "being a video" and "receiving many likes" are independent.
3. c) Calculate the probability that the post receives many likes knowing that it is not a video.

Let $A$, $B$ and $C$ be three events of the same random experiment such that:

• $A$ and $B$ are independent.

• $A$ and $C$ are mutually exclusive.

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

Calculate in detail by applying the properties of probability:

1. a) $P(A \cup B)$
2. b) $P(A \cup C)$
3. c) $P(\bar{A} \cap \bar{B})$
4. d) $P((A \cup B) \cap C)$ knowing also that $B$ and $C$ are independent.

Consider a random experiment where the sample space is $E$. Let $S_1$ and $S_2$ be two events such that their probabilities depend on an unknown value $k \in \mathbb{R}$. It is known that:

• $P(S_1) = k$

• $P(S_2) = k^2 + 0,1$

• $P(S_1 \cup S_2) = 0,7$

Determine the value of $k$ knowing that the events $S_1$ and $S_2$ are mutually exclusive. Once $k$ is found, check if it would be possible for $S_1$ and $S_2$ to be independent under the same individual probability conditions (ignoring the previous union). Justify whether there is any value of $k$ that allows independence and the union simultaneously.

A cybersecurity company has installed two intrusion detection systems, $D_1$ and $D_2$, which operate independently. The probability that, in the event of an external attack, system $D_1$ triggers the alarm is $0,92$. On the other hand, the probability that system $D_2$ does not detect the attack is $0,15$.

If an unexpected attack occurs, calculate:

1. a) The probability that at least one of the two systems detects the attack.
2. b) The probability that only system $D_2$ triggers the alarm.
3. c) Knowing that the attack has been detected by at least one of the systems, what is the probability that both systems have functioned correctly?

An 11th grade project team is made up of 5 girls and 3 boys. The teacher must choose 2 people at random to present the project. Calculate:

1. a) The sample space of the experiment (based on the gender of those chosen).
2. b) The probability that the two chosen are of the same gender.
3. c) The probability that at least one of those chosen is a girl.

In a safe there are two compartments. Compartment A contains 4 $\$10$ bills and 6 $\$20$ bills. Compartment B contains 7 $\$10$ bills and 3 $\$20$ bills. A die is rolled: if a number less than 3 is rolled, compartment A is chosen, and otherwise compartment B. Then, a bill is drawn at random.

1. a) Draw the corresponding tree diagram.
2. b) What is the probability that the drawn bill is $\$20$?
3. c) If the drawn bill is $\$10,$ what is the probability that it comes from compartment A?

In an e-sports tournament, a team has to pass three consecutive knockout stages to win the prize. The probabilities of passing each stage are dependent on success in the previous one due to accumulated fatigue:

• The probability of passing Stage 1 is $P(F_1) = 0,8$.

• If they pass Stage 1, the probability of passing Stage 2 is $P(F_2/F_1) = 0,7$.

• If they do not pass Stage 1, they are automatically eliminated (obviously there is no Stage 2).

• If they have passed the first two stages, the probability of winning Stage 3 is $P(F_3 / F_1 \cap F_2) = 0,5$.

Calculate:

1. a) The probability that the team is eliminated exactly in Stage 2.
2. b) The probability that the team wins the tournament (passes all three stages).
3. c) Knowing that the team has not won the tournament, what is the probability that they did not even pass the first stage?

A study has been carried out on the use of sunglasses and visual fatigue in drivers. Out of 200 drivers, 120 use sunglasses regularly. Of those who use sunglasses, $10\%$ claim to have visual fatigue after a long trip, while among those who do not use them, the percentage of fatigue rises to $35\%$.

1. a) Organize the information in a contingency table.
2. b) If we choose a driver at random, what is the probability that they have visual fatigue?
3. c) If a driver claims not to have visual fatigue, what is the probability that they use sunglasses?

A diagnostic laboratory analyzes water samples to detect a specific bacterium. It is known that $5\%$ of the sources in a region are contaminated ($C$). The test used has a probability of giving a false positive (testing positive when there is no bacterium) of $2\%$, and a probability of giving a false negative (testing negative when there is a bacterium) of $1\%$.

1. a) Construct the tree diagram corresponding to this situation.
2. b) Calculate the probability that the test is positive ($+$).
3. c) If the test of a source has been positive, what is the probability that it is actually contaminated? Comment on the result obtained from the point of view of the test's reliability.