Exercises: The binomial distribution

In a popular strategy board game, a player rolls a special four-sided (tetrahedral) die numbered from 1 to 4. The random variable $X$ is defined as the "score obtained on the die".

1. a) Create the probability distribution table for the variable $X$.
2. b) Calculate the expected value or mean $\mu$ and interpret its meaning in the context of the game.

A café near the high school has analyzed 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 logically whether the following random experiments can be modelled by a binomial distribution. If so, state their parameters $n$ and $p$:

1. a) Toss a coin 15 times and record the number of heads obtained.
2. b) Draw, one by one and without replacement, 3 cards from a standard deck of cards and count the number of diamonds.
3. c) Ask 40 people chosen at random in a shopping center if they have seen the latest superhero movie, knowing that $30\%$ of the population has already seen it.

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

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

It is known that $10\%$ of the messages an influencer receives on their social media accounts are negative comments (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 comments.

In an electronic components factory, the probability of a circuit being 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.

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 do not answer any question correctly?
2. b) Using the complementary event, calculate the probability that they answer at least one question correctly.

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 ($k = 0, 1, 2, 3, 4$). Show that the sum of all the probabilities is equal to 1.

A recent study on digital habits states that $35 \%$ of American teenagers aged 15 to 16 prefer to use podcasts as their main source of entertainment over conventional social networks. If we select a random group of 10 students in 11th grade:

1. a) Justify why this experiment can be modeled by 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 the standard deviation of the number of students who prefer podcasts in this group.

In a game of a well-known board game, a player needs to roll at least one "6" when rolling 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" is rolled in the 5 rolls.

1. a) Determine the probability function of the variable $X$.
2. b) Calculate the probability that the player is not eliminated (that is, they roll at least one six).
3. c) Represent the distribution graphically using a bar chart.

A courier company uses electric vans for its urban deliveries. It is known from experience that the probability of a van needing an additional recharge 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 the additional recharge.
2. b) What is the probability that more than 6 vans complete their shift without needing an extra recharge? (Be careful to correctly define what you consider a "success").

A chain of "vintage" clothing stores has found that $20 \%$ of the customers who enter their establishment make a purchase of more than $\$50$. If 12 customers enter independently in one morning:

1. a) What is the probability that the number of customers who spend 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" (junk mail). 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 there is exactly one "spam" message in a block.
3. c) If the 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 sports center, $75 \%$ of the users enrolled in swimming classes are able to swim 100 feet 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 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, it is necessary to answer at least 5 questions correctly. What is the probability that the student passes purely by chance?
3. c) What is the most probable score (mode) that the student will obtain?

It has been observed that $5 \%$ of the LED bulbs of a certain brand leave the 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 bulbs in a box?
2. b) If the hotel returns the box if it finds more than one defective 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 bulbs? (Use the probability calculated for a single box).

A content creator on a well-known social network knows that the probability of a follower sharing one of their videos is exactly $p = 0,02$. If the creator publishes a series of videos and wants to be sure, with a probability greater than $95\%$, that at least one of their videos is shared by a specific follower who watches all their content:

1. a) Determine the minimum number of videos, $n$, that they must publish to achieve this goal.

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 assembly line for e-sports components, 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 a defect).

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

A courier company uses drones to deliver parcels in rural areas. The probability of a drone successfully completing a flight is $p$. The company has observed that the probability that, on a day with $4$ scheduled flights, all of them are successful is exactly twice the probability that only $3$ of them are.

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

1. b) With the value of $p$ obtained, if $6$ flights are scheduled one day, what is the probability that at least $2$ flights fail?

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 $P(X=0) = 0,0001$ and the probability of obtaining $n$ successes is $P(X=n) = 0,01$.

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 $(n+1)p - 1 \le k \le (n+1)p$.

Two e-sports players, Alex and Bea, compete in a target-shooting minigame. Alex has a probability of hitting the target of $70\%$, while Bea has a probability of $60\%$. If each of them takes $5$ independent shots:

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

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

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

A six-sided die is rolled $n$ times. Let $X$ be the random variable that expresses the number of times a multiple of $3$ is obtained. If it is known that the probability of not obtaining 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:

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 represent by $P(X=k)$ the probability that exactly $k$ friends cancel, justify logically, without calculating all the values, what the value of $k$ is that maximizes this probability.