Counting Techniques

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'?

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?

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).

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 $4321$).

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.

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:

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 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.

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 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.

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?

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?

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?

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 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?

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?