Exercises: Concepts

Classify the following experiments as random or deterministic, briefly justifying your answer in each case:

1. a) Observing whether the next bus passing the stop in front of the school is route 5.
2. b) Measuring the time it takes for an object to fall to the ground from a height of 2 metres in a vacuum.
3. c) Predicting the winner of the next Christmas Lottery draw.
4. d) Calculating the cost of 3 kg of apples if we know the price is $1,50 \text{ €/kg}$.

We are playing with a spinner divided into eight equal sectors, numbered from 1 to 8. Write the sample space $E$ for this experiment and list the elements of the following events:

1. a) $A =$ "Obtaining a number greater than 5".
2. b) $B =$ "Obtaining an even number".
3. c) $C =$ "Obtaining a multiple of 3".

In a sports bag there are 4 red T-shirts, 3 blue ones and 1 white one. If we draw a T-shirt without looking, indicate which of the following options best describes the probability of each event (use the terms: very likely, unlikely, impossible, certain):

1. a) Drawing a green T-shirt.
2. b) Drawing a T-shirt that is not white.
3. c) Drawing a red, blue or white T-shirt.
4. d) Drawing a blue T-shirt.

Two coins are tossed into the air and the result of each one is recorded (C for Heads, X for Tails).

1. a) Use a tree diagram to find all the possible outcomes.
2. b) Write the sample space of the experiment.
3. c) What is the elementary event "getting two heads"?

We consider the random experiment of choosing a letter at random from the word MATEMATICAS. Define the following concepts associated with this experiment:

1. a) The sample space $E$.
2. b) The event $V =$ "Choosing a vowel".
3. c) The event $C =$ "Choosing a consonant".
4. d) Check whether $V$ and $C$ are complementary events.

A die has its faces painted in colours: 3 faces are red, 2 are green and 1 is yellow. If we roll the die:

1. a) What is the probability of "getting red"? Express the result as a fraction.
2. b) What is the probability of "not getting red"?
3. c) What is the technical term for the event in part b) relative to that in part a)?

One card is drawn at random from a Spanish deck of 40 cards. Given the events $A =$ "Drawing a King" and $B =$ "Drawing a card of Coins", state:

1. a) Are $A$ and $B$ compatible or incompatible events? Why?
2. b) What is the intersection event $A \cap B$?
3. c) Describe in words the union event $A \cup B$.

The figure shows how the deck is organized: 4 suits and 10 cards per suit (numbered from 1 to 7, and the face cards 10 = jack, 11 = knight and 12 = king).

In a lottery drum there are balls numbered from 1 to 20. If we draw a ball at random, classify the following events as elementary or compound:

1. a) $S_1 =$ {obtaining the number 13}.
2. b) $S_2 =$ {obtaining a prime number less than 10}.
3. c) $S_3 =$ {obtaining a multiple of 7}.
4. d) $S_4 =$ {obtaining the largest even number in the bag}.

In Maths class, Marcos analyzes the random experiment of rolling a fair six-sided die with faces numbered from 1 to 6. After thinking for a moment, he states: "Let's consider the events $A = \text{“rolling a number less than or equal to 4”}$ and $B = \text{“rolling a number greater than or equal to 3”}$. They are complementary events because, when united, they cover all possible outcomes: $A \cup B = \{1, 2, 3, 4, 5, 6\} = E$".\n\na) Explain in detail what the error in Marcos's reasoning is. What two conditions must be met simultaneously for two events to be formally complementary?\nb) Write by extension (listing its elements) the true complement of $A$, denoted by $\bar{A}$.\nc) Find the intersection $A \cap B$ and deduce from it whether $A$ and $B$ are compatible or mutually exclusive events.

In a board game, a numbered token is drawn at random from a bag. The sample space of the experiment is $E = \{1, 2, 3, 4, 5, 6, 7, 8\}$. You are asked to construct and determine the elements of two events, $A$ and $B$, that simultaneously satisfy the following four conditions:\n1. Event $A$ must be formed by exactly 5 elementary events.\n2. The intersection of both events must be $A \cap B = \{2, 6\}$.\n3. The union of both events must contain all elements of the sample space except 8, that is, $A \cup B = E \setminus \{8\}$.\n4. Event $B$ has exactly 4 elements.\n\na) Write the elements that make up event $A$ and event $B$, justifying step by step how you decide the placement of each number.\nb) Write the complement of $B$, $\bar{B}$, and determine whether $A$ and $\bar{B}$ are compatible or mutually exclusive events.

On your mobile phone you have 3 social media apps, 2 game apps and 1 music app installed. When you receive a notification, it can come from any of these 6 apps with the same probability.

1. a) Define the sample space $E$ of this random experiment.
2. b) Describe the event $A$ = "The notification is not from a social media app".
3. c) Write an impossible event related to this context.

A survey is carried out among the 30 students in a class about their streaming preferences. It is known that 18 students watch Netflix ($N$), 12 watch Disney+ ($D$) and 5 students watch both platforms. If we choose a student at random:

1. a) Use a Venn diagram to represent the situation.

1. b) Calculate the probability that the student watches only Netflix.

1. c) Calculate the probability that the student watches neither of the two platforms (the complement of the union).

In a fantasy trading card game, a deck contains the following cards: 8 "Warrior" type, 6 "Wizard" type, 4 "Elf" type and 2 "Dragon" type. If you draw a card at random, calculate the probability that:

1. a) It is a "Wizard".
2. b) It is a "Warrior" or an "Elf".
3. c) It is not a "Dragon".

A content creator has rolled an 8-sided die (octahedron) 500 times to check whether it is loaded. The results obtained for the face with the number "8" are shown in the following table:

1. a) Calculate the relative frequencies for each stage of the experiment.

1. b) What value does the probability of getting an "8" seem to stabilise around?

1. c) If the die were perfect (fair), what should its theoretical probability be according to Laplace's rule?

At a gaming convention, 200 attendees were classified according to their age and preferred console, obtaining the following contingency table:

If we select an attendee at random, find the probability that they:

1. a) Prefer Nintendo.
2. b) Are over 14 years old.
3. c) Are under 14 years old and prefer PlayStation.
4. d) Are under 14 years old, given that they prefer Nintendo.

To go to a festival, Lucía has in her wardrobe 3 sweatshirts (black, white and red) and 2 pairs of trousers (jeans and sweatpants). She chooses a sweatshirt and a pair of trousers at random, with all options equally likely.

1. a) Draw the tree diagram that represents all the possible outfit combinations (sweatshirts and trousers).
2. b) What is the probability that Lucía chooses the red sweatshirt?
3. c) What is the probability that she does not choose the sweatpants?

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

We consider the random experiment of rolling two cubic dice and adding their scores. Let the events be:

• $A$ = "The sum is an even number".

• $B$ = "The sum is a multiple of 5".

1. a) Calculate $P(A)$ and $P(B)$.
2. b) Are the events $A$ and $B$ mutually exclusive or not? Justify your answer by finding the elements of $A \cap B$.
3. c) Calculate the probability of the union $P(A \cup B)$.

At a tech fair there is a spinner divided into 12 equal sectors numbered from 1 to 12. The game consists of spinning the spinner and observing the winning number.

1. a) Find the probability of obtaining a prime number.
2. b) Find the probability of obtaining a number greater than 4 and less than 9.
3. c) If we win a prize if a multiple of 3 or an odd number comes up, what is the probability of winning?

To prepare the activities for the high school mathematics week, the sample space of two different experiments is to be analysed:\n- Experiment 1: Simultaneously roll two 6-sided cubic dice and calculate the positive difference between the higher and the lower score (if they are equal, the difference is 0).\n- Experiment 2: Randomly answer a 4-question quick quiz where each question only allows two possible answers: True ($V$) or False ($F$).\n\na) Compare the two usual tools for describing compound experiments: the two-way table and the tree diagram. Justify which of the two is more suitable for Experiment 1 and which for Experiment 2, explaining why a two-way table is not practical for Experiment 2.\nb) Apply the multiplication rule ($m \cdot n \dots$) to calculate how many possible outcomes make up the sample space of each of the two experiments.\nc) Using the method you considered ideal for Experiment 1, represent all possible outcomes and describe by listing its elements the event $D = \text{“the difference between the faces is greater than or equal to 4”}$.

A metropolitan bus company in Spain classifies each vehicle arriving at the station according to two independent observation variables: engine technology, which can be Electric ($E$) or Hybrid ($H$), and service punctuality, which can be On time ($P$), with Slight delay ($R$), or Cancelled ($C$). The arrival of the next service is observed.\n\na) Justify why observing the next service constitutes a random and not a deterministic experiment.\nb) Explicitly write the sample space $\Omega$ as a set of ordered pairs. How many elementary events does it have?\nc) Define by listing its elements (writing all its elements between braces) the events:\n - $A = \text{“the bus is hybrid”}$.\n - $B = \text{“the bus experiences an incident (slight delay or cancellation)”}$.\nd) Determine the sets of outcomes that form the events $A \cap B$, $A \cup B$, and the complementary event $\bar{B}$. Explain what event $\bar{B}$ means in practice for a person waiting at the bus stop.

A group of friends creates a security key for their video game server. The key must have 3 distinct characters chosen from the letters $\{A, B, C\}$ and the numbers $\{1, 2, 3\}$.

1. a) Describe the sample space $\Omega$ if the first character must be a letter and the last character must be a number.

1. b) If a key is chosen at random from all possible keys (without the restrictions of the previous part, but keeping the condition that the 3 characters are distinct), what is the probability that the key consists only of numbers?

In a secondary school in Madrid, $60\%$ of 3rd ESO students use the platform Twitch, $50\%$ use TikTok, and $20\%$ do not use either of them. If we choose a student at random, calculate the probability of the following events:

1. a) That they use both platforms.

1. b) That they use TikTok but not Twitch.

1. c) Are the events "Using Twitch" and "Using TikTok" mutually exclusive? Justify your answer using the probability values.

Two popular streamers, Alba and Brais, are going to do a collaboration. They have an urn with 4 physical challenge cards and 6 mental challenge cards. They decide to draw two cards successively without replacement.

1. a) Draw a tree diagram representing all the possibilities and assign the probability to each branch.

1. b) Calculate the probability that the second card drawn is a mental challenge, knowing that the first was physical.

1. c) What is the probability that they perform one challenge of each type?

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

A delivery drone must land on a square plot of land with side length $10$ metres. Inside the plot there is a circular safety zone (where it must not land) with a radius of $2$ metres, and an optimal landing zone which is a small central square with side length $1$ metre.

Assuming that the drone lands at any point on the plot with equal probability:

1. a) What is the probability that the drone lands in the optimal landing zone?

1. b) What is the probability that it lands outside the circular safety zone? (Use $\pi \approx 3,14$).

In an esports tournament, the peripherals used by 120 participants are analysed. It is known that 75 use a mechanical keyboard and the rest use a membrane keyboard. Of those using a mechanical keyboard, 45 also use a wireless mouse. In total, 65 participants use a wireless mouse.

1. a) Organise these data in a contingency table.

1. b) If we choose a participant at random, what is the probability that they use a membrane keyboard and a wired mouse?

1. c) If we know that a participant uses a wireless mouse, what is the probability that their keyboard is a membrane keyboard?

We roll a six-sided die that we suspect is loaded. After rolling it 500 times, we have obtained the following frequencies: the number 6 came up 125 times, the number 1 came up 35 times, and the numbers 2, 3, 4, and 5 came up with equal frequency among themselves.

1. a) Calculate the relative frequency of the event "Rolling a 6".

1. b) According to the Law of Large Numbers, what probability would you assign to the event "Rolling a number less than 6"?

1. c) If the die were fair, what would be the theoretical probability of rolling a 6? Compare both results.

A board game consists of flipping a fair coin (heads and tails are equally likely) and, depending on the result, doing the following:

• If it lands heads, a fair 4-sided die (numbered from 1 to 4) is rolled.

• If it lands tails, a ball is drawn from an urn containing 2 red balls and 3 green balls.

1. a) Represent the experiment using a tree diagram, indicating the probability of each branch.

1. b) Calculate the probability of getting an even number on the die if the coin landed heads.

1. c) What is the total probability that the final result of the experiment is a red ball?

In a tennis tournament final between Carlos and Rafa, the first one to win 3 sets wins. The tournament is interrupted by rain when Carlos is leading 2 sets to 1. Knowing that both have the same probability of winning each remaining set:

1. a) Use a tree diagram to calculate each player's probability of winning the tournament had play continued.

1. b) If the total prize is $12.000$ €, how should it be fairly divided according to each player's probability of winning at the moment of the interruption?

Three 3rd-year ESO students (Álex, Bea, and Carlos) organise a chess tournament during the breaks on Wednesday, Thursday, and Friday. To decide who referees each day, they place three slips of paper with their names into an urn and draw one at random each morning, returning it to the urn so that everyone has the same chance again the next day.\n\na) Estimate: Before calculating or listing cases, intuitively compare these two events:\n - Event $M = \text{“the same person referees all three days”}$.\n - Event $D = \text{“different people referee all three days”}$.\n Reason, without doing numerical calculations yet, which of the two events you think will have more favourable outcomes in the sample space and why.\nb) Model the experiment using a tree diagram or the multiplication rule to determine the total number of elementary events in the sample space $E$.\nc) Exhaustively list all the elementary events that form event $M$ and all those that form event $D$.\nd) Check: Count the favourable outcomes of $M$ and of $D$. Assess whether the result matches your initial estimate and explain mathematically why the number of ways to arrange three different people ($3 \cdot 2 \cdot 1$) is much greater than repeating the same person ($3 \cdot 1 \cdot 1$).

An automatic AEMET weather station at a mountain pass records two variables at midnight to summarise the day:\n- Precipitation type: Snow ($N$), Rain ($L$), or Dry/No precipitation ($S$).\n- Mean temperature: Below zero ($B$) or Zero or positive ($C$).\nIn a purely combinatorial model, $3 \cdot 2 = 6$ possible outcomes would be obtained. However, the physical conditions of the location introduce a real restriction: it is physically impossible for it to snow if the temperature is zero or positive, so the outcome $(N, C)$ can never occur.\n\na) Write the real sample space $E$ of the random experiment taking into account this physical restriction. How many elementary events make it up?\nb) List the elements of the following events:\n - $A = \text{“precipitation is recorded (snow or rain)”}$.\n - $B = \text{“the mean temperature is below zero”}$.\nc) Determine the elements of $A \cap B$, $A \cup B$, and the event $\overline{A \cup B}$ (complement of the union).\nd) Justify whether the events “the temperature is below zero with no precipitation” and “it is a snowy day” are compatible or mutually exclusive using the mathematical definition of intersection.\ne) An operator claims: «Since for it to snow the temperature must be below zero, the event “snowing” and the event “below zero temperature” are equal events». Explain with set-algebraic rigour why this statement is false, using the concept of event inclusion.