Events

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 in a training session and observing whether the ball goes into the goal.
2. b) Measuring the time taken for an object to fall from a height of 5 meters in a vacuum.
3. c) Observing the number of WhatsApp messages you receive between 17:00 and 18:00 on any given Monday.
4. d) Calculating the total cost of buying 3 kilograms of oranges if the price per kilogram is $\$25$.
5. e) Drawing a card from a Spanish deck and observing whether 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 recorded.

Based on this experiment:

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

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

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

A group of friends analyse their hobbies. Let $D$ be the event "playing sport" and $M$ be the event "playing a musical instrument". Express the following situations symbolically using operations on events:

1. a) The friend plays sport or plays an instrument (or both).
2. b) The friend plays sport and also plays an instrument.
3. c) The friend plays sport but does not play any instrument.
4. d) The friend neither plays sport nor plays any instrument.

In a Year 12 class, 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 at least one 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 total in the class, calculate the probability that the chosen student studies German.

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) $A - B$ (outcomes that are in $A$ but not in $B$)

Reasonably prove, using De Morgan's laws and the axiomatic properties of probability, the following equality for any two events $A$ and $B$:

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

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

Consider the experiment of drawing a card from a Spanish deck of 40 cards. State 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 events $A$ and $B$ within a sample space $E$.

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

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

In a survey carried out among 1st-year Bachillerato 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.

1. b) The probability that they are subscribed to Netflix but not to HBO Max ($P(N - H)$).

1. c) Represent the situation using a Venn diagram (draw the sample space $E$ and the sets $N$ and $H$), indicating the probabilities of each region.

Given two events $A$ and $B$ in 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 $B$ incompatible events? Justify your answer.
2. b) Are $A$ and $B$ independent events?
3. c) Calculate $P(A \cap \bar{B})$.

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

Using the data from the chart, 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.

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 given that it is not a video.

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 randomly chosen participant 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?

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.

A Year 12 project team consists 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 (considering the gender of those chosen).
2. b) The probability that both people chosen are of the same gender.
3. c) The probability that at least one of those chosen is a girl.

A study has been carried out on the use of sunglasses and eye strain in drivers. Out of 200 drivers, 120 wear sunglasses regularly. Of those who wear sunglasses, $10\%$ state that they suffer from eye strain after a long journey, whereas among those who do not wear them, the percentage suffering from eye strain rises to $35\%$.

1. a) Organise the information in a contingency table.
2. b) If we choose a driver at random, what is the probability that they suffer from eye strain?
3. c) If a driver states that they do not suffer from eye strain, what is the probability that they wear sunglasses?

In a safe there are two compartments. Compartment A contains 4 10 pesos notes and 6 20 pesos notes. Compartment B contains 7 10 pesos notes and 3 20 pesos notes. A die is rolled: if a number less than 3 is obtained, compartment A is chosen; otherwise, compartment B is chosen. Then, a note is drawn at random.

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

You can do it on paper and send a photo with your answer.

A diagnostic laboratory analyzes water samples to detect a specific bacterium. It is known that $5\%$ of the springs 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 bacterium) of $1\%$.

1. a) Construct the tree diagram corresponding to this situation.

1. b) Calculate the probability that the test result is positive ($+$).

1. c) If the test of a spring has come out positive, what is the probability that it is actually contaminated? Comment on the result obtained from the point of view of the reliability of the test.

You can do it on paper and send a photo with your answer.

In a study on digital media consumption among teenagers, two events are analysed: $A$ (the user regularly uses a microblogging platform) and $B$ (the user consumes content on video streaming platforms). After conducting 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, giving reasons, the following probabilities:

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.

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 incompatible. Once $k$ has been found, check whether 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 exists 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 work independently. The probability that, in the event of an external attack, system $D_1$ raises the alarm is $0.92$. On the other hand, the probability that system $D_2$ does not detect the attack is $0.15$.

The diagram shows the possible branches (✓ = detects, ✗ = does not detect).

If an unexpected attack occurs, calculate:

1. a) The probability that at least one of the two systems detects the attack.

1. b) The probability that only system $D_2$ raises the alarm.

1. c) Given that the attack has been detected by at least one of the systems, what is the probability that both systems worked correctly?

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

• $A$ and $B$ are independent.

• $A$ and $C$ are incompatible.

• $P(A) = 0.4$; $P(B) = 0.5$ and $P(C) = 0.2$.

Calculate in detail, 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.

In an e-sports tournament, a team has to pass three consecutive knockout stages to win the prize. The probabilities of passing each stage depend on success in the previous stage 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 in exactly Stage 2.
2. b) The probability that the team wins the tournament (passes all three stages).
3. c) Given that the team did not win the tournament, what is the probability that they did not even pass the first stage?