Exercises: Mean and standard deviation of the BIN

A recent study on digital habits indicates that $85\%$ of American teenagers between 15 and 16 years old 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 proven that the probability of a user "skipping" a song recommended by the algorithm before it finishes 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 about the dispersion of the data in this context.

In a vocational training center, $40\%$ of the students use public transportation to get to class. A random sample of $50$ students from the center is taken. Calculate the mean and standard deviation of the number of students who use public transportation 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 mean number of baskets she is expected to score?
2. b) Calculate the standard deviation. What would happen to the standard deviation if her probability of success were $50\%$? Show 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 smartphone components factory, it is known that $2\%$ of the touch screens are produced with a minor cosmetic defect. The screens are sent to stores in boxes of $250$ units.

1. a) Calculate the mean number of defective screens we will find 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" appears 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 with three possible answers each, of which only one is correct. If a student answers all the 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 well-known social network, it has been estimated that the probability of a user interacting with a cultural content post 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 $\$50$ plus $\$2$ 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, Ashley and Brandon, practice free throws. Ashley has a probability of scoring of $0,8$ and takes $50$ shots. Brandon has a probability of $0,6$ but takes $80$ shots.

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

1. b) Which of the two players shows a 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 parcels in urban centers. The probability of a drone suffering a delay due to weather conditions is $5\%$. If $200$ independent flights are made in a day:

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

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

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

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

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 vary with respect to the original sample? Justify your answer with the calculation.

It has been proven that $40\%$ of the 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 $X$ that defines the number of students who take the exam.

1. b) In binomial distributions, it is considered that the most "typical" or probable values are found in the interval $(\mu - \sigma, \mu + \sigma)$. Calculate this interval for this specific case.

The quality control department of an LED bulb factory knows that the probability of a unit being defective is $p = 0,02$. The bulbs are packed in boxes of $n$ units.

1. a) If it is desired that the mean number of defective bulbs per box is exactly $1,6$, how many bulbs must each box contain?

1. b) With that value of $n$, calculate the probability that there are no defective bulbs in a box.

Let $X$ be a random variable that follows 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 (i.e., success is less probable than failure), which value of $p$ would you choose? Justify your answer.

A content creator on a social network knows that the probability of a new user subscribing to their channel after watching a "short" is $p = 0,12$. The algorithm informs them that, in their last campaign, the mean number of new subscribers was $18$ people.

1. a) Determine the sample size $n$ (number of views) that the algorithm has analyzed.
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 to deliver parcels in smart cities. It is known that the variable $X$, which defines the number of failed deliveries in a batch of $n$ shipments, follows a binomial distribution. Following an internal study, it has been proven 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 a single 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 fail?

A cybersecurity analyst is studying phishing attacks. The probability of an employee clicking 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 the email must be sent to 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 standard deviation of the number of students who go to school by electric scooter in groups of $50$ people is $\sigma = 3$.

1. a) Find the two possible values of the probability $p$ that a student uses an electric scooter.
2. b) If it is known that the scooter is a minority mode of transportation ($p < 0,5$), 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 twice 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 flipped $n$ 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 tails.
2. b) The number of flips $n$ made.
3. c) The probability of getting exactly $2$ heads in only $5$ flips.

In a quality control check at an LED bulb factory, the number of defective units in boxes of a certain size is analyzed. The variance of defective bulbs per box is $3,75$ and it is known that the probability of a bulb being defective is $5\%$.

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

Let $X$ be a random variable that follows 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 any success in a sample of only $3$ individuals from that same population.