The binomial distribution

A coffee shop near the school has analysed the number of coffees each customer orders in a morning. The probability distribution of the variable $X$ = "number of coffees per customer" is given by the following table:

1. a) Find the value of $k$ so that it is a valid probability distribution.
2. b) What is the probability that a customer orders at least 2 coffees?

Determine with reasons whether the following random experiments can be modelled using a binomial distribution. If so, state their parameters $n$ and $p$:

1. a) Tossing a coin 15 times and recording the number of heads obtained.
2. b) Drawing 3 cards one by one without replacement from a Spanish deck and counting the number of coins.
3. c) Asking 40 randomly selected people in a shopping centre if they have seen the latest superhero movie, knowing that $30\%$ of the population has already seen it.

An archer has a probability of $80\%$ of hitting the bullseye. If they perform a round of 10 independent shots, calculate the parameters of the corresponding binomial distribution:

1. a) Expected number of bullseyes (mean $\mu$).
2. b) Standard deviation $\sigma$.

A recent study on digital habits claims that $35 \%$ of Spanish teenagers aged between 15 and 16 prefer using podcasts as their main source of entertainment over conventional social media. If we select a random group of 10 students from 1st Bachillerato:

1. a) Justify why this experiment can be modelled using a binomial distribution and define its parameters $n$ and $p$.
2. b) Calculate the probability that exactly 4 of them prefer podcasts.
3. c) Find the mean and standard deviation of the number of students who prefer podcasts in this group.

In a round of a board game, a player needs to roll at least one "6" when throwing a fair six-sided die in 5 attempts to avoid being eliminated. Let $X$ be the random variable that counts the number of times the number "6" comes up in the 5 throws.

1. a) Determine the probability function of the variable $X$ (calculate $P(X=k)$ for $k=0,1,2,3,4,5$).

1. b) Calculate the probability that the player is not eliminated (that is, that they get at least one six).

1. c) Graph the distribution using a bar chart (on the horizontal axis, the number of sixes $k$; on the vertical axis, $P(X=k)$).

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

A courier company uses electric vans for its urban deliveries. Experience shows that the probability of a van requiring an additional charge before finishing its shift is $0,15$. If the company has a fleet of 8 vans operating simultaneously:

1. a) Calculate the probability that, on a given day, at most 2 vans need an additional charge.
2. b) What is the probability that more than 6 vans complete their shift without needing an extra charge? (Be careful to correctly define what you consider a "success").

In a popular strategy board game, a player rolls a fair special four-sided die (tetrahedral), with faces numbered from 1 to 4, so that all of them have the same probability of coming up. The random variable $X$ is defined as "score obtained on the die".

1. a) Build the probability distribution table of the variable $X$.

1. b) Calculate the expected value or mean $\mu$ and interpret its meaning in the context of the game.

A vintage clothing chain store has found that $20 \%$ of customers who enter its shop make a purchase over $50$ €. If 12 customers enter independently one morning:

1. a) What is the probability that the number of customers spending more than $50$ € is equal to the mean of the distribution?
2. b) Calculate the probability that at least 3 customers make such a purchase.

An email server detects that $10 \%$ of the messages it receives are "spam" (unsolicited email). The security filter analyses the messages in blocks of 20.

1. a) Find the expected number of "spam" emails per block and its standard deviation.
2. b) Calculate the probability that a block contains exactly one "spam" message.
3. c) If a block is marked as "suspicious" when it contains more than 3 "spam" messages, what is the probability that a block is marked as such?

In a school archery club, the probability of an archer hitting the target on a shot is $p$. It has been observed that the probability of hitting the target on all shots in a series of $4$ shots is exactly double the probability of hitting the target on only $3$ of them.

1. a) Calculate the value of $p$.

1. b) Using the value of $p$ obtained, if she takes a series of $6$ shots, what is the probability that she misses at least $2$ of them?

A content creator on a well-known social network knows that the probability that a follower shares any one of their videos is exactly $p = 0,02$. Assume that each video is shared or not by that follower independently of the others, so that the number of shared videos, $X$, follows a binomial distribution $B(n;\,0,02)$.

The creator publishes a series of $n$ videos and wants to be sure, with a probability greater than $95\%$, that at least one of them is shared by a particular follower who watches all of their content.

1. a) Determine the minimum number of videos, $n$, that they must publish to achieve this goal, that is, so that $P(X \geq 1) > 0,95$.

1. b) If they finally publish $n = 150$ videos, calculate the expected value and the standard deviation of the number of videos shared by that follower. Interpret the meaning of the expected value in this context.

On an e-sports component assembly line, it has been determined that the probability of a microchip being defective is $p$. It is known that in a batch of $n$ microchips, the mean number of defective components is $12$ and the variance is $10,56$.

1. a) Calculate the values of the parameters $n$ (batch size) and $p$ (probability of defect).

1. b) If a random sample of $10$ microchips is drawn from a new batch, what is the probability that exactly $2$ are defective? And the probability that at most one is defective?