Exercises: Counting techniques

In an electronic sports (eSports) club, the specialization of its 100 members is analyzed. It is known that 65 of them regularly play League of Legends (event $L$), 40 play Valorant (event $V$) and 15 play both video games.

Calculate the probability that, when a member is chosen at random:

1. a) They play at least one of the two video games ($L \cup V$).

1. b) They play neither of the two.

1. c) They play only League of Legends.

Illustrate the situation with a Venn diagram. Here is the starting outline, which you must complete with the number of members in each region:

To improve the security of a mobile app, a developer decides to create access codes consisting of an uppercase letter (from the standard 26-letter alphabet) followed by three digits (from 0 to 9).

• How many different codes can be created if digits can be repeated?

• How many different codes would there be if the three digits must be distinct from one another?

• If a code with repeatable digits is generated at random, what is the probability that the letter is a 'Z'?

An urn contains 12 glass balls for an optics experiment: 5 are blue, 4 are red, and 3 are yellow. If we draw a ball at random, calculate the following probabilities using Laplace's rule:

• That the ball is red or yellow.

• That the ball is not blue.

• That the ball is white.

In a robotics laboratory, 8 proximity sensors in perfect working order and 2 defective sensors have been mixed together in a box. A student draws two sensors consecutively and without replacement (without returning the first one to the box).

Calculate the probability that:

• Both sensors work correctly.

• The first is defective and the second is in good working order.

A survey was conducted among the 60 2nd Baccalaureate students at a technological secondary school regarding their use of social media to keep up with science news. The results are shown in the following contingency table:

	Use X (Twitter)	Use Instagram	Total
Male students	18	12	30
Female students	10	20	30
Total	28	32	60

If we select a student at random, calculate the probability that:

• The student is female.

• They use Instagram to stay informed.

• Given that the student is male, they prefer to stay informed via the social network X.

A student has two exams on the same day: Mathematics and English. Based on their past marks, they estimate that the probability of passing Mathematics ($M$) is $0,7$, passing English ($I$) is $0,8$, and passing both is $0,6$.

• Are the events "Passing Mathematics" and "Passing English" independent? Justify your answer mathematically.

• Calculate the probability that the student passes at least one of the two exams.

In a hospital laboratory, two automatic analyzers, $A$ and $B$, process blood samples. Analyzer $A$ processes $60 \%$ of the samples and $B$ processes the remaining $40 \%$. It is known that $2 \%$ of the samples from analyzer $A$ must be repeated due to a reading error, while for $B$ that percentage is $5 \%$.

Draw the corresponding tree diagram and calculate the probability that a randomly chosen sample must be repeated.

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

An engineering class has to choose a committee of 3 representatives. If there are 20 students in the class, calculate:

• In how many ways can the committee be chosen if the three roles (class representative, deputy representative, and secretary) are distinct?

• In how many ways can it be chosen if the three representatives have equal rank and order does not matter?

• If there are 12 men and 8 women, what is the probability that the committee (without distinct roles) consists entirely of women? (Express the fraction using combinations).

In a secondary school, the student delegate council for 2nd year of Bachillerato consists of 10 students: 6 girls and 4 boys. To form a coexistence committee, a group of 3 delegates must be chosen by lottery.

Marcos wants to calculate the probability that the committee consists of exactly 2 girls and 1 boy, and reasons as follows:

"For the first girl there are 6 options, for the second there are 5 options, and for the boy there are 4 options. Therefore, the favorable outcomes are $6 \cdot 5 \cdot 4 = 120$. Since the possible outcomes of choosing 3 people in order out of 10 are $10 \cdot 9 \cdot 8 = 720$, the required probability is $\frac{120}{720} = \frac{1}{6}$".

1. a) Accurately identify Marcos's conceptual error when calculating the favorable outcomes and explain why his approach distorts the calculation of the sample space.
2. b) Solve the problem correctly using combinations $\binom{n}{r}$ by Laplace's rule and compare your result with the one obtained by Marcos.

In a group of 15 students in 2nd year of Bachillerato taking Mathematics II, there are three inseparable friends: Lucía, Mateo, and Sofía. The department will randomly select a team of 4 students to represent the school in the local university's Mathematics Olympiad.

1. a) Before carrying out any mathematical calculations, make a qualitative and intuitive estimate: do you think the probability that at least two of the three friends are together in the 4-person team is greater than $50\,\%$, around $15\,\%-20\,\%$, or is it a very unlikely event (less than $5\,\%$)? Briefly justify the intuition on which you rely.
2. b) Model the problem using combinatorial techniques and calculate the exact value of this probability as an irreducible fraction and as a percentage.
3. c) Compare the obtained value with your initial estimate and argue why human intuition about group coincidences often fails in this type of draw.

To improve the security of an esports platform, users are required to create a 6-character password. The first two characters must be uppercase letters of the Spanish alphabet (27 letters) and the remaining four must be digits from $0$ to $9$.

Calculate how many different passwords can be generated if:

1. a) Repetitions of letters and numbers are allowed.
2. b) Repeating letters is not allowed, but repeating numbers is allowed.
3. c) The letters must be distinct and the numbers must form a strictly increasing sequence (for example: $1258$, but neither $1225$ nor $4321$).

In a performance study of a new graphics processor, the events $A$: "the processor exceeds $100$ FPS on ultra settings" and $B$: "the processor maintains a temperature below $70^\circ$C" are defined. The following probabilities are known:

$P(A) = 0,65$

$P(B) = 0,50$

$P(A^c \cup B^c) = 0,70$

Find, with reasoning:

1. a) The probability that both events occur simultaneously.
2. b) The probability that only event $A$ occurs.
3. c) Are the events $A$ and $B$ independent? Justify your answer.

A robotics laboratory has two boxes of components. Box 1 contains $8$ proximity sensors and $4$ actuators. Box 2 contains $5$ proximity sensors and $7$ actuators. A box is chosen at random and two components are drawn successively without replacement.

1. a) Calculate the probability that both components drawn are actuators.
2. b) Given that both components drawn were proximity sensors, what is the probability that they came from Box 1?

A software development company uses three servers to host its API: server $S_1$ handles $45\%$ of the traffic, $S_2$ handles $35\%$, and $S_3$ handles the rest. Due to high load, server $S_1$ fails (error 500) in $2\%$ of requests, $S_2$ in $3\%$, and $S_3$ in $5\%$.

1. a) If a user makes a request, what is the probability that the server returns an error?
2. b) If a request has failed, what is the probability that it was handled by server $S_3$?

A telecommunications engineer estimates that the probability of a data packet being lost during a wireless transmission under interference conditions is $0,15$. A burst of $10$ packets is sent independently.

1. a) What is the probability that exactly $3$ packets are lost?
2. b) Calculate the probability that at least $2$ packets are lost.
3. c) What is the expected number of lost packets and their standard deviation?

The compilation time of a complex engineering project follows a normal distribution with a mean of $120$ seconds and a standard deviation of $15$ seconds.

1. a) Calculate the probability that the project compiles in under $100$ seconds.
2. b) Calculate the probability that the compilation time is between $110$ and $140$ seconds.
3. c) What compilation time is exceeded by only $5\%$ of the slowest processes?

In a technical certification exam, scores follow a $N(70, 12)$ distribution. The institution decides to award an "Excellence Commendation" to students who achieve a mark in the $90$ percentile.

1. a) What is the minimum mark required to receive this commendation?
2. b) If $250$ students take the exam, how many are expected to receive the commendation?
3. c) If we choose two students at random, what is the probability that both receive the commendation?

A social network detects that $10\%$ of created accounts are "bots". If $400$ new accounts are registered in one hour:

1. a) Explain why the number of bots can be approximated using a normal distribution.
2. b) Using the normal approximation with continuity correction, calculate the probability that there are between $35$ and $50$ bots (inclusive).
3. c) Calculate the probability that the number of bots is greater than $45$.

A rural health center in Castilla y León has a staff of 11 emergency professionals, including family doctors and nursing staff. To organize the weekend night shift, a duty team of 3 professionals is randomly selected from the total of 11.

It is known with certainty that the probability that the duty team consists of 2 doctors and 1 nurse is exactly twice the probability that it consists of 1 doctor and 2 nurses.

1. a) If we denote by $m$ the number of doctors on staff (with $2 \le m \le 9$), write the combinatorial equation that models the given condition using Laplace's rule.
2. b) Solve this equation algebraically to find the exact composition of the staff (how many doctors and how many nurses there are at the center).
3. c) With the staff now determined, calculate the probability that the night shift is composed exclusively of professionals from the same healthcare group.

The standard vehicle registration system of the Directorate-General for Traffic (DGT) in Spain assigns to each car a code consisting of 4 digits (from 0000 to 9999) followed by a block of 3 consonant letters from the Spanish alphabet, having excluded the 5 vowels (A, E, I, O, U) to avoid acronyms and offensive words, as well as the letters Ñ and Q due to possible visual confusion. In this way, a set of 20 allowable consonants is available.

1. a) Model the assignment of license plates and calculate how many different vehicles can be registered in total. Justify the counting techniques applied to each block of the license plate.
2. b) A traffic control camera records an offender's license plate, but the image is partially blurry: the officer verifies that the numerical block is a four-digit palindrome with different first and second digits (for example, 7337), and that the three letters are all distinct and appear in strict ascending alphabetical order (for example, CFX). How many license plates in the DGT's theoretical vehicle fleet satisfy this pattern?
3. c) If a vehicle is chosen at random from among all possible ones in the system, determine the probability that its license plate contains no repeated digit and also contains no repeated consonant. Express the result as an irreducible fraction and to four decimal places.

A cybersecurity company generates access keys by permuting the characters of the word BLOCKCHAIN. One of these keys is chosen at random.

1. a) Calculate the probability that the chosen key starts and ends with a vowel.
2. b) Determine the probability that all the vowels appear together in the key.
3. c) Given that the key starts with B, what is the probability that the second letter is an L?

A data storage server receives requests from three types of clients: Bronze ($50\%$), Silver ($30\%$) and Gold ($20\%$). Due to saturation, the server may reject requests. It is known that the probability of a request being rejected is $10\%$ for Bronze clients, $5\%$ for Silver clients and $1\%$ for Gold clients.

1. a) If a request is selected at random and turns out to be rejected, what is the probability that it comes from a Bronze client?
2. b) Following a system update, the overall probability of rejection has dropped to $4\%$. If the probabilities for Silver and Gold clients remain the same, what is the new probability of rejection for Bronze clients?

An urn $A$ contains 4 white balls and 6 black balls, and an urn $B$ contains 7 white balls and 3 black balls. A fair die is rolled: if a number less than or equal to 2 is obtained, a ball is drawn from urn $A$ and placed into urn $B$, and then a ball is drawn from $B$. If a number greater than 2 is obtained, a ball is drawn from urn $B$ and placed into urn $A$, and then a ball is drawn from $A$.

Calculate the probability that the second ball drawn is white.

A laser aiming system for eye surgery has a deviation error $X$, measured in micrometres, which follows a probability density function given by:

$$f(x) = \begin{cases} k(4x - x^3) & \text{if } 0 \le x \le 2 \\ 0 & \text{otherwise} \end{cases}$$

1. a) Determine the value of the constant $k$ so that $f(x)$ is a valid probability density function.
2. b) Calculate the probability that the deviation is greater than 1 micrometre.
3. c) Obtain the expected value of the deviation. What does this value mean in the context of the surgery?

A satellite communication system transmits data packets. Each packet has an independent probability $p = 0,002$ of becoming corrupted during transmission. To ensure reception, a block of $2500$ packets is sent.

1. a) Justify why the number of corrupted packets can be approximated by a normal distribution.
2. b) Calculate the probability that between 4 and 8 packets (inclusive) are corrupted.
3. c) If the error correction system can only repair up to 10 packets per block, what is the probability that the block is unrepairable?

The response time of an industrial sensor follows a normal distribution $N(\mu, \sigma)$. It is known that $15,87\%$ of response times take more than $82$ milliseconds, and that only $2,28\%$ of response times are faster than $58$ milliseconds.

1. a) Calculate the parameters $\mu$ and $\sigma$ of the response time.
2. b) If the manufacturer guarantees that the sensor responds in less than $85$ ms, what percentage of sensors will not satisfy the guarantee?

In an engineering exam taken by 100 students, the marks follow a normal distribution with mean $6,2$ and standard deviation $1,5$. The lecturer decides that the top $10\%$ of students with the highest marks will receive an "Outstanding" grade, whilst those whose mark is more than one standard deviation below the mean will fail.

1. a) What is the minimum mark required to obtain an "Outstanding"?
2. b) How many students are expected to fail the exam?
3. c) If we choose three students at random, what is the probability that at least one of them has a mark above 7?

An electronic device consists of two critical components, $C_1$ and $C_2$, connected in parallel (the device functions if at least one of the two functions). The probability that $C_1$ fails within a year is $0,15$, and that $C_2$ fails is $0,20$. However, the failures are not independent: if $C_1$ fails, the probability that $C_2$ also fails increases to $0,60$.

1. a) Calculate the probability that the device fails within a year.
2. b) Are the events "$C_1$ fails" and "$C_2$ fails" dependent? Justify your answer with calculations.
3. c) If the device has failed, what is the probability that component $C_1$ has failed?

In a cybersecurity technical audit, 18 data servers belonging to a banking entity are analyzed. It has been detected that 4 of them have an unpatched critical security flaw, while the remaining 14 are completely secure. The audit team randomly selects a sample of 5 servers to subject them to a penetration test. We want to calculate the probability of event $A$: "there are at least two vulnerable servers in the selected sample".

1. a) Carry out the calculation of $P(A)$ using two different analytical approaches:

• Approach 1 (Direct counting): partition the favorable outcomes according to the exact number of vulnerable servers found.

• Approach 2 (Complementary event): calculate the probability of the complementary event $A^c$.

Verify that both procedures lead to the same irreducible fraction.

1. b) If the batch size increased to 30 servers and the sample to be selected were 10 servers, analyze which of the two methods offers a real operational advantage against calculation errors and justify it based on the combinatorial structure of the partitions.
2. c) A team assistant proposes calculating the favorable outcomes of event $A$ as follows: "We first choose 2 vulnerable servers from the 4 available in $\binom{4}{2}$ ways; then, we choose the remaining 3 servers from the 16 remaining in the batch in $\binom{16}{3}$ ways. Therefore, the favorable outcomes are $\binom{4}{2} \cdot \binom{16}{3}$". Analytically demonstrate why this reasoning is flawed and specify whether it produces an overestimation or an underestimation of the actual number of outcomes.

To form an examination board for teaching civil service examinations in an autonomous community, the administration calls 5 board members by strict lottery from a closed list of 30 qualified applicants, consisting of 18 women and 12 men. Public service regulations require the principle of balanced representation: the board cannot have more than $60\,\%$ of members of the same sex (that is, the board must have at least 2 women and at least 2 men).

1. a) Model the selection of the board using combinations and calculate the probability that a board drawn at random directly complies with the balanced representation regulations.
2. b) Suppose that the list of 30 people includes two specific teachers (a man and a woman) who are first-degree relatives, so administrative law prohibits both from being on the same board. If the administration rules out in advance through the draw the possibility of both coinciding, restricting the sample space solely to boards where the two relatives do not coincide, determine whether the probability that the board meets parity increases or decreases compared to that obtained in section a). Quantitatively justify your conclusion.
3. c) As an alternative, it is proposed to modify the draw procedure: instead of a single draw of 5 people from the 30 candidates, use two separate ballot boxes (one for women and one for men) and randomly draw 3 women and 2 men with probability $0,5$, or 2 women and 3 men with probability $0,5$. Mathematically justify whether this new mechanism maintains or alters the individual probability of each teacher being selected compared to the original equiprobable draw.