Mean and standard deviation of the binomial distribution

A recent study on digital habits indicates that $85\%$ of Spanish 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.

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.

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.

It has been verified that $40\%$ of students at a language centre sit 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 sitting 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.

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 $50$ € 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.

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.

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.

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.