Exercises: Mean and standard deviation of the BIN

A recent study on digital habits indicates that $85\%$ of Mexican teenagers aged 15 to 16 use two-factor authentication on their video game accounts. If we select a random sample of $20$ teenagers:

1. a) Identify the parameters $n$ and $p$ of the binomial distribution that models this situation.
2. b) Calculate the expected number (mean) of teenagers who use this security system in the sample.

A music streaming platform has verified that the probability of a user "skipping" an algorithm-recommended song before it ends is $0.3$. If one afternoon you listen to a playlist of $12$ recommended songs:

1. a) Determine the mean and variance of the number of songs you are expected to skip.
2. b) Calculate the standard deviation and interpret what information it provides regarding data dispersion in this context.

In a vocational training college, $40\%$ of the students use public transport to get to class. A random sample of $50$ students from the college is taken. Calculate the mean and standard deviation of the number of students who use public transport in this sample.

A professional basketball player has a free-throw success rate of $80\%$. If she takes a series of $15$ shots in a training session:

1. a) What is the expected mean number of baskets she will score?
2. b) Calculate the standard deviation. What would happen to the standard deviation if her probability of scoring were $50\%$? Verify this by calculation.

Consider a discrete random variable $X$ that follows a binomial distribution $B(n, p)$. If we know that the number of trials is $n = 100$ and that the mean of the distribution is $\mu = 60$:

1. a) Find the value of the probability of success $p$.
2. b) Calculate the variance of this distribution.

In a factory manufacturing smartphone components, it is known that $2\%$ of the touchscreens have a small cosmetic defect. The screens are sent to shops in boxes of $250$ units.

1. a) Calculate the mean number of defective screens found in a box.
2. b) Calculate the standard deviation of the variable "number of defective screens per box".

A fair six-sided die is rolled a total of $30$ times, and the number of times the number "6" comes up is recorded.

1. a) Define the random variable and its distribution.
2. b) Calculate the expected value of the variable and its standard deviation. Round the results to two decimal places.

A multiple-choice exam consists of $10$ questions, each with three possible answers, of which only one is correct. If a student answers all questions at random:

1. a) What is the mean number of correct answers?
2. b) If the student needs at least $5$ correct answers to pass, is the mean value above or below the pass mark? Justify your answer.

On a popular social network, it has been estimated that the probability of a user interacting with a post containing cultural content is $p = 0.18$. If an influencer launches a campaign that reaches a group of $n = 150$ followers:

1. a) Calculate the expected number of followers who will interact with the post and its standard deviation.

1. b) If the cost of the campaign is a fixed $1\,000$ pesos plus $40$ pesos for each expected interaction (based on the mean), what is the expected budget for this advertising action?

A discrete random variable follows a binomial distribution $X \sim B(n, p)$. It is known that its mean is $\mu = 12$ and its standard deviation is $\sigma = 3$.

1. a) Determine the values of the parameters $n$ (number of trials) and $p$ (probability of success).

1. b) Justify whether this distribution could be approximated by a normal distribution, based on the values obtained.

Two basketball players, Alba and Bruno, practice free throws. Alba has a scoring probability of $0.8$ and shoots $50$ throws. Bruno has a probability of $0.6$ but shoots $80$ throws.

1. a) Which of the two players has a higher expected number of baskets?

1. b) Which of the two players shows greater dispersion (standard deviation) in their results? Interpret what this means in terms of consistency in the game.

A logistics company uses drones to deliver packages in urban areas. The probability that a drone suffers a delay due to weather conditions is $5\%$. If $200$ independent flights are conducted on a given day:

1. a) Calculate the value of the mean and variance of the number of flights that arrive late.

1. b) The maintenance protocol is activated if on a given day the number of delays exceeds the value $\mu + 2\sigma$. Calculate this threshold value.

In a survey on reading habits conducted with $60$ teenagers, it is known that the expected number of regular digital press readers is $21$.

1. a) Find the probability $p$ that a randomly chosen teenager is a regular digital press reader.

1. b) Calculate the standard deviation of this distribution.

1. c) If we select another sample where the number of teenagers is doubled ($n=120$), how does the standard deviation change relative to the original sample? Justify your answer with calculations.

It has been verified that $40\%$ of students at a language center take the official B2 level exam. If there are $100$ students enrolled this year:

1. a) Determine the parameters of the variable $X$ defining the number of students taking the exam.

1. b) In binomial distributions, the most "typical" or likely values are considered to lie within the interval $(\mu - \sigma, \mu + \sigma)$. Calculate this interval for this specific case.

The quality control department of an LED light bulb factory knows that the probability of a unit being defective is $p = 0.02$. The light bulbs are packaged in boxes of $n$ units.

1. a) If it is desired that the mean number of defective light bulbs per box is exactly $1.6$, how many light bulbs should each box contain?

1. b) With that value of $n$, calculate the probability that a box contains no defective light bulbs.

Let $X$ be a random variable following a binomial distribution. If the number of trials is $n = 25$ and the variance is $Var(X) = 4$:

1. a) Calculate the two possible values of the probability of success $p$.

1. b) For each value of $p$ found, calculate its corresponding mean $\mu$.

1. c) If we know that the distribution is positively skewed (that is, success is less likely than failure), which value of $p$ would you choose? Justify your answer.

A social media content creator knows that the probability that a new user subscribes to their channel after watching a "short" is $p = 0.12$. The algorithm informs them that, in their latest campaign, the mean number of new subscribers was $18$ people.

1. a) Determine the sample size $n$ (number of views) analysed by the algorithm.
2. b) Calculate the standard deviation of this distribution.
3. c) What is the probability that the number of subscribers is exactly equal to the mean?

A logistics company uses drones for package delivery in smart cities. It is known that the variable $X$ defining the number of failed deliveries in a batch of $n$ shipments follows a binomial distribution. After an internal study, it has been verified that the mean number of failures is $4$ and the variance is $3.2$.

1. a) Calculate the parameters $n$ and $p$ of the distribution.
2. b) If an individual shipment is made, what is the probability that the delivery is successful?
3. c) What is the probability that in a batch of $n$ shipments, none of them fail?

A cybersecurity analyst is studying phishing attacks. The probability that an employee clicks on a malicious link is $0.005$. The analyst wants to send a test campaign to a group of employees of size $n$.

Calculate the minimum number of employees to whom the email must be sent so that the probability of at least one of them clicking on the link is greater than $90\%$.

In a study on the use of sustainable transport, it has been observed that the number of students who come to school by electric scooter, in groups of $50$ people, follows a binomial distribution $B(50,\,p)$, where $p$ is the probability that a student uses an electric scooter. The standard deviation of that number of users is $\sigma = 3$.

The figure shows how the standard deviation $\sigma=\sqrt{50\,p\,(1-p)}$ varies as a function of $p$, along with the line $\sigma = 3$.

1. a) Find the two possible values of the probability $p$ that a student uses an electric scooter.

1. b) If it is known that the scooter is a minority means of transport ($p < 0.5$), calculate the mean number of users in a sample of $100$ students.

A strategy video game has a probability $p$ of obtaining a "legendary item" when opening a chest. A player opens $200$ chests and observes that the standard deviation of their reward is double what it would be if the probability of success were $0.01$.

1. a) Set up an equation to find the value of $p$.
2. b) Determine the expected mean number of legendary items in the $200$ chests with that $p$.

A biased coin is tossed $n$ times. It is known that the probability of getting heads is $p$. If the mean number of heads obtained is $15$ and the standard deviation is $\sqrt{10}$, calculate:

1. a) The probability $p$ of getting heads.

1. b) The number of tosses $n$ performed.

1. c) The probability of getting exactly $2$ heads in only $5$ tosses.

In a quality control check at an LED light bulb factory, the number of defective units in boxes of a given size is analysed. The variance of defective light bulbs per box is $3.75$ and it is known that the probability of a light bulb being defective is $5\%$.

1. a) How many light bulbs does each box contain?
2. b) If the manufacturer wants to halve the standard deviation, how should the number of light bulbs per box change while maintaining the same manufacturing quality ($p$)?

Let $X$ be a random variable following a binomial distribution $B(n, p)$. It is known that its mean is $\mu = 30$.

1. a) What is the minimum value its standard deviation can take? Justify your answer by reasoning about the possible values of $p$.
2. b) If the standard deviation is $\sigma = 5$, calculate the probability of obtaining at least one success in a sample of only $3$ individuals from that same population.