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 modeled 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 center if they have seen the latest superhero movie, knowing that $30\%$ of the population has already seen it.

In an electronic components factory, the probability that a circuit is defective is $0.02$. If they are packed in boxes of 5 units, calculate the probability that there are no defective circuits in a box.

Given a binomial distribution $X \sim B(4; 0.3)$, construct its complete probability distribution table by calculating the probabilities for all possible values of the variable ($k = 0, 1, 2, 3, 4$). Check that the sum of all probabilities is equal to 1.

In a sports center, $75 \%$ of the users enrolled in swimming lessons are able to swim 100 meters continuously without resting. If we choose 6 users at random:

1. a) What is the probability that all of them are able to swim that distance?
2. b) What is the probability that none of them are able to?
3. c) Calculate the probability that half of them (exactly 3) meet the requirement.

A multiple-choice exam consists of 8 questions with 4 options each, of which only one is correct. If a student answers all the questions at random:

1. a) What is the probability that they get no questions correct?
2. b) Using the complementary event, calculate the probability that they get at least one question correct.

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 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.

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 $\sigma$.

It is known that $10\%$ of the messages an influencer receives on their social media are negative reviews (the rest are positive or neutral comments). If they have received 6 new messages today:

1. a) Define the random variable and its distribution.
2. b) Calculate the probability that exactly 2 of those messages are negative reviews.

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.

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?

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").

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?

It has been observed that $5 \%$ of the LED light bulbs of a certain brand are factory defective. A hotel buys boxes of 15 units to renew the lighting in its corridors.

1. a) What is the probability that there are no defective light bulbs in a box?
2. b) If the hotel returns the box when it finds more than one defective light bulb, what is the probability that a box is returned?
3. c) If 100 boxes are bought, in how many of them is it expected to find exactly 2 defective light bulbs? (Use the probability calculated for a single box).

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 multiple-choice exam consists of 10 questions with 4 answer options each, of which only one is correct. A student who has not studied at all decides to answer all the questions at random.

1. a) Determine the probability that the student answers exactly 5 questions correctly.
2. b) To pass the exam, a student needs to answer at least 5 questions correctly. What is the probability that the student passes by pure chance?
3. c) What is the most likely mark (mode) that the student will obtain?

For a discrete random variable $X$ that follows a binomial distribution $B(n, p)$, it is known that the probability of obtaining zero successes is $P(X=0) = 0.027$ and the probability of obtaining $n$ successes is $P(X=n) = 0.343$.

1. a) Using the properties of powers, find the value of $n$ and that of $p$.

1. b) Calculate the probability of obtaining a number of successes equal to the mode of the distribution. Hint: The mode in a binomial distribution is the integer value $k$ such that $(n+1)p - 1 \le k \le (n+1)p$.

A competitive exam consists of $80$ multiple-choice questions with $4$ answer options each, of which only one is correct. A student decides to answer at random $12$ questions they do not know how to answer. Let $X$ be the variable counting the number of correct answers out of those $12$.

1. a) Calculate the interval $[\mu - \sigma, \mu + \sigma]$ for this variable.

1. b) Determine the probability that the number of correct answers lies outside this interval. Comment on whether the result obtained is consistent with the expected dispersion in a distribution of this type.

Two e-sports players, Alex and Bea, compete in an aiming mini-game. Alex has a hit probability of $70\%$, while Bea has a probability of $60\%$. If each of them takes $5$ shots independently:

1. a) What is the probability that Alex gets more hits than Bea?

1. b) What is the probability that between them they get a total of exactly $9$ hits?

A six-sided die is rolled $n$ times. Let $X$ be the random variable expressing the number of times a multiple of $3$ is obtained. If it is known that the probability of not getting any multiple of $3$ in the $n$ rolls is $16/81$:

1. a) Calculate the number of rolls $n$.

1. b) For that value of $n$, calculate the probability of obtaining a multiple of $3$ at least twice, given that it has appeared at least once.

A streaming platform knows that $25\%$ of its users cancel their subscription before the third month. In a group of $8$ friends who have subscribed simultaneously (assume that each one decides independently of the others):

1. a) Calculate the probability that the number of friends who cancel their subscription is greater than the mean plus one standard deviation.

1. b) If we denote by $P(X=k)$ the probability that exactly $k$ friends cancel, justify with reasoning, without calculating all the values, which value of $k$ maximizes this probability.