Exercises: Counting techniques

A streaming platform lets you customize your user profile by choosing an avatar from 4 different animals (lion, panda, cat and wolf) and a background color from 3 options (blue, green and orange).

1. a) Describe the sample space $E$ of the possible profile combinations.

1. b) How many elements does the sample space have? Use the multiplication principle to justify it.

In your school cafeteria menu you can choose a starter between salad, pasta or rice; a main course between chicken or fish; and a dessert between fruit or yoghurt.

1. a) Represent all possible complete menu options using a tree diagram.
2. b) If you choose a menu at random, what is the probability that it includes "pasta" as a starter?

A combination padlock for a locker has a 3-digit code. Each digit can be a number from $1$ to $5$.

1. a) How many different combinations can be created if numbers can be repeated?
2. b) How many combinations are there if the three numbers must be distinct?

In a 1st Baccalaureate class there are 30 students. It is known that 18 of them passed the last Mathematics exam and 12 passed the English exam. If there are 5 students who passed both subjects:

1. a) Calculate the probability that, choosing a student at random, they passed Mathematics or English.
2. b) Calculate the probability that they passed neither of the two subjects.

Five friends (Ana, Beto, Clara, David and Elena) go to the cinema and sit in a row of five consecutive seats.

1. a) In how many different ways can they sit in the row?
2. b) If they decide that Ana and Beto must always sit at the two ends of the row, how many different arrangements are possible now?

A ball is drawn from an urn containing 10 balls numbered from $1$ to $10$. We consider the following events:

$A = \text{"Getting a prime number"}$

$B = \text{"Getting a multiple of 3"}$

1. a) Write down the elements forming events $A \cup B$ and $A \cap B$.
2. b) Calculate the probabilities $P(A)$, $P(B)$ and $P(A \cup B)$ using Laplace's rule.

A band has prepared 10 songs for their new album, but the record label tells them they can only include 4 of them.

1. a) How many different selections of 4 songs can they make if the order does not matter in which they are chosen?
2. b) What are these types of groupings called in combinatorics? Justify your answer.

In a local chess tournament 8 players participate. Three different trophies will be awarded for first, second and third place.

1. a) In how many different ways can the three prizes be awarded?
2. b) If last year's champion participates again and we are certain he will finish in the top three, how many podium finishes are possible now?

In a 1st year Bachillerato class of 20 students, a committee of 3 people must be formed to coordinate the school's cultural week. All members of the committee will have the same responsibility and there will be no hierarchical roles.

Lucas calculates the number of possible committees as follows:

"For the first person there are 20 options; for the second, 19; and for the third, 18. Therefore, the total number of possible committees is $20 \cdot 19 \cdot 18 = 6840$".

1. a) Identify the conceptual error Lucas makes and explain in detail why his calculation counts duplicate committees.
2. b) Correct Lucas's calculation by applying the appropriate counting technique to determine the actual number of distinct committees that can be formed.
3. c) If the school management decided to assign the three people three clearly differentiated roles (event coordination, treasury, and communication), would Lucas's original reasoning be correct then? Justify your answer by distinguishing the importance of order in each of the two situations.

In the 1st year Bachillerato Contemporary World History exam, the test consists of a bank of 6 essay questions, and each student must freely choose 4 of them to answer.

1. a) Before carrying out any calculation using combinatorial formulas, make an intuitive and reasoned estimate of how many distinct choices of 4 questions are possible: do you think the result will be between 1 and 10, between 11 and 30, or will it exceed 50 options? Briefly justify what you base your initial forecast on.
2. b) Model the situation and formally calculate the exact number of options using the relevant counting technique, justifying why the order of choice does or does not matter.
3. c) Check whether your initial estimate was correct and explain, using the symmetry property of combinatorial numbers $\binom{n}{k} = \binom{n}{n-k}$, why selecting 4 questions to answer out of 6 available produces exactly the same number of options as selecting 2 questions to discard.

A group of 12 1st Baccalaureate students has been selected to attend a conference on sustainability. The school must choose 4 of them to form a committee of spokespersons.

• a) In how many different ways can the committee be chosen if no distinction is made between roles?

• b) If there are 7 girls and 5 boys among the 12 students, what is the probability that the committee consists of exactly 2 boys and 2 girls?

A streaming platform uses a security code for parental control. The code consists of 2 letters (from a 26-letter alphabet, without Ñ) followed by 3 digits (from 0 to 9).

• a) How many different codes can be created if letters and numbers can be repeated?

• b) If a user forgets their code but remembers that it begins with the letter "A" and ends with the number "7", how many possible combinations would they have to try?

• c) If the code were generated at random without allowing any repeated characters (letters or numbers), what would be the probability of the code being "BX123"?

In a local basketball league, the effectiveness of a player's three-point shots has been analyzed. It is known that if he scores the first three-pointer, the probability of scoring the second is $0,7$. However, if he misses the first, the probability of scoring the second is only $0,4$. It is known that the probability of scoring the first three-pointer of a game is $0,5$.

1. a) Draw the tree diagram associated with the experiment of taking two consecutive three-pointers.

1. b) Calculate the probability that the player scores exactly one of the two shots.

1. c) If we know that the player missed the second shot, what is the probability that he had scored the first?

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

A tech company manufactures wireless headphones in two plants, one in Spain and the other in Vietnam. The plant in Spain produces $40\%$ of the total and Vietnam the rest. It has been found that $3\%$ of the headphones from Spain have connection faults, whereas in the Vietnam plant this fault appears in $6\%$ of production.

• a) If we buy a pair of headphones at random, what is the probability that they work correctly?

• b) If we buy a pair of headphones and they turn out to have connection faults, what is the probability that they were manufactured in the plant in Spain?

A survey was conducted among 200 young people about their reading and television viewing habits. The data are shown in the following contingency table:

If a young person is chosen at random from this survey, calculate:

• a) The probability that they read regularly and watch television less than 2 hours a day.

• b) Given that the young person reads regularly, what is the probability that they watch television more than 2 hours a day?

• c) Are the events "Reads regularly" and "Watches television more than 2 hours a day" independent? Justify your answer mathematically.

In a toolbox there are 5 Phillips screwdrivers and 3 flat-head screwdrivers. 2 screwdrivers are chosen at random, one after the other, without putting the first one back into the box.

1. a) Calculate the probability that both are of the same type.

1. b) Calculate the probability that the second one is flat-head given that the first one was Phillips.

1. c) If the experiment were done with replacement, how would the probability in part a) change? Calculate its new value and compare it with the one in part a).

You can use a tree diagram that you draw yourself.

Let $A$ and $B$ be two events in a sample space such that their probabilities are known: $P(A) = 0,6$, $P(B) = 0,4$ and $P(A \cap B) = 0,2$.

Calculate the following probabilities using the properties of probability:

• a) $P(A \cup B)$

• b) $P(\bar{A} \cap \bar{B})$ (Probability that neither A nor B occurs).

• c) $P(A | \bar{B})$ (Probability of A conditioned on B not occurring).

A fashion designer is creating a capsule collection. For the final catwalk show, they have to arrange 6 models in a single line.

• a) In how many different ways can the 6 models walk down the catwalk?

• b) If two of the models are sisters and the designer insists that they must walk immediately next to each other, in how many ways can the line be arranged now?

• c) If the designer decides that the two sisters cannot be together under any circumstances, how many options are there?

To organize a youth volunteer campaign in a municipality, a working group made up of 4 students chosen by lot from a volunteer group of 12 people consisting of 7 girls and 5 boys is created. It is desired to determine how many different ways the group can be formed so that it has at least one girl.

Two members of the management team propose different counting strategies:

• Method A (direct counting): Count separately the possible committees with exactly 1 girl, with 2 girls, with 3 girls, and with 4 girls, and add the four results.

• Method B (complementary method): Calculate the total possible groups of 4 people without any restriction and subtract those groups in which no girl participates.

1. a) Develop and numerically solve the problem step by step following Method A.
2. b) Develop and numerically solve the problem step by step following Method B, verifying that both paths lead to the same solution.
3. c) Compare both methods and justify which of the two is more advantageous and efficient if the volunteer group were to increase to 20 girls and 15 boys and an 8-person delegation with at least one girl had to be chosen.

The regular vehicle registration system in Spain, regulated by the Directorate-General for Traffic (DGT), assigns each vehicle a license plate formed by two consecutive blocks: first, four numerical digits (from $0000$ to $9999$) and, then, three letters of the alphabet.

To avoid rude words or confusing acronyms, as well as reading errors with numbers, Spanish regulations exclude from the letter block the five vowels (A, E, I, O, U) and the consonants Ñ and Q. Thus, exactly 20 distinct consonants are used. Both in the digits and in the letters, repeating characters is allowed (for example, the license plate $0044\text{--}BBD$ is completely valid).

1. a) Model the space of all possible license plates by identifying the appropriate counting technique for each of the two blocks. Determine the total theoretical capacity of vehicles that can be registered with this system in Spain.
2. b) According to the reports on the Spanish vehicle fleet from the National Statistics Institute (INE) and the DGT, around 33 million registered vehicles currently circulate in Spain. Based on the result obtained in the previous section, reason whether this system is at risk of exhaustion in the short or medium term.
3. c) If the DGT approved a technical reform prohibiting the repetition of any letter within the same license plate (allowing, for example, BCD but not BBC), how many valid license plates would remain and what exact percentage of license plates compared to the original system would be lost? Justify the combinatorial calculations.

An IT security system requires passwords to have exactly $8$ characters. These must be chosen from the $26$ letters of the alphabet (case-insensitive) and the $10$ numerical digits ($0-9$). A password is considered "strong" if it contains at least two numerical digits and does not repeat any character. If a program generates a random password that meets the condition of not repeating characters, calculate the probability that it is "strong".

In an inter-school debate tournament, $12$ students participate: $7$ girls and $5$ boys. For the final stage, a team of $4$ representatives must be chosen at random. Calculate the probability that the chosen team satisfies both conditions simultaneously:

• There is representation of both sexes.

• The number of girls is strictly greater than the number of boys.

A logistics company organizes its routes using a barcode of $10$ vertical stripes. Each stripe can be black, grey or white. For scanner readability reasons, codes with three consecutive stripes of the same colour are not allowed.

1. How many different codes of $10$ stripes can be formed in total (without restrictions)?

2. If a code is chosen at random from all possible codes (without restrictions), what is the probability that the stripes in the odd positions are all black and those in the even positions are all white or grey?

A data analyst studies user behaviour on a streaming platform. They have the following information about a sample of $500$ users:

• $65\%$ are under $25$ years old.

• $40\%$ of the users use the platform mainly to watch documentaries.

• Among those who do not watch documentaries, $70\%$ are under $25$ years old.

If a user is selected at random and turns out to be under $25$ years old, calculate the probability that their main preference is documentaries. Use a contingency table to organize the information.

In a toolbox there are $15$ LED light bulbs, $3$ of which are known to be defective. If we take out $4$ light bulbs one by one without replacement:

1. Calculate the probability that the fourth light bulb taken out is the second defective one to appear in the process.

2. If the selections were made with replacement, would the previous probability change? Justify your answer using counting techniques.

It is known that in a certain city it rains on $20\%$ of days ($LL$), it is very windy on $10\%$ ($V$) and both conditions occur simultaneously on $5\%$.

1. Calculate the probability that on a randomly chosen day it does not rain and is not windy.

2. Given that it did not rain on a certain day, what is the probability that it was very windy?

3. Are the events "raining" and "being very windy" independent? Prove it mathematically.

A multiple-choice exam consists of $5$ questions with $4$ answer options each, of which only one is correct. A student who has not studied at all decides to answer at random. However, they have a strategy: for each question, they discard one of the options that they know for sure is false (as it is absurd) and choose at random from the remaining three.

1. Calculate the probability that they get exactly $3$ correct answers.

2. What is the probability that they get at least one question right?

An urn contains $n$ white balls and $4$ red balls. Two balls are drawn simultaneously (or successively without replacement). If it is known that the probability that both balls are white is equal to $1/3$, determine the total number of balls in the urn. (Note: you must set up a combinatorial equation and solve it to find the value of $n$).

In the customer service department of an e-commerce company, 8 women and an unknown number $h$ of men ($h \ge 2$) work. On the occasion of an internal quality evaluation, management is going to randomly select a pair of 2 workers from the department to coordinate the audits, with no distinction of roles between them.

It is known with certainty that the probability that the two people selected by lot are of the same sex is exactly $\frac{7}{15}$.

1. a) Express in terms of $h$, using combinatorial numbers and justifying why these are combinations without repetition, both the total number of possible pairs that can be formed in the department and the number of favorable pairs formed by two people of the same sex.
2. b) Set up the equation associated with the indicated probability and solve it algebraically step by step to find the possible values of the number of men $h$.
3. c) Critically interpret the results obtained: how is it possible that there are two distinct whole numbers of men that provide exactly the same probability of obtaining a pair of the same sex? Explain this phenomenon by analyzing how the favorable cases and the possible cases vary together when moving from one solution to the other.

In the Provincial Court, in accordance with the Organic Law on the Jury Court in Spain, a citizens' jury must be formed consisting of 9 regular jurors and 2 alternate jurors. Once the challenge phase is complete, 15 preselected candidates remain in the room: 9 women and 6 men. The appointment process is carried out by rigorous drawing in two consecutive phases: first, the 9 regular jurors are randomly drawn (among whom there is no order or hierarchy); immediately afterward, from among the unselected candidates, the 2 alternate jurors are drawn in order, assigning to the first one drawn the status of "first alternate" and to the second that of "second alternate" (here order is relevant to determine priority in the event of a replacement).

1. a) Model, using the multiplication principle and the relevant combinatorial techniques, the total number of distinct configurations in which the complete panel (9 regular jurors and 2 ordered alternates) can be formed.
2. b) If the draw is equiprobable, calculate the probability that the panel of 9 regular jurors is formed with a female majority (that is, with at least 5 female regular jurors). Detail and justify the count of favorable cases by applying combinatorics.
3. c) During the judicial proceedings, a lawyer raises the following procedural objection: "The order in which the draws are carried out alters the possibilities: if we had first drawn the 2 alternates with their order of precedence from among the 15 candidates, and from the remaining 13 we had drawn the 9 regular jurors, the total number of possible configurations of the panel would have been completely different". Prove algebraically whether the lawyer's statement is true or false by comparing the combinatorial expressions of both procedures.