Exercises: The binomial distribution

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 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 $\sigma$.

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.

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.

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.

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 study on youth mobility in a regional capital, it is known that $70\,\%$ of Bachillerato students regularly use the public transport pass ($p = 0,7$). A random sample of $n = 5$ students from this group is selected. To calculate the probability that exactly 4 of them use this pass, Marcos performs the following operation in his notebook:

In view of his result, Marcos concludes: "It is an extremely rare event, barely a $7,2\,\%$ probability, which shows that we will almost never find 4 users in groups of 5 people."

1. a) Reasonably identify the conceptual and calculation error that Marcos made when applying the binomial distribution. What essential element has he omitted and what combinatorial meaning does it have in this real context?
2. b) Correct the calculation by determining the exact value of $P(X = 4)$ and explain why the new value aligns much more logically with the intuition of the problem.

According to official statistics on sustainability and waste management in Spain, $80\,\%$ of households sort lightweight packaging into the corresponding container ($p = 0,8$). A team of environmental inspectors randomly selects 4 independent households in a town.

1. a) Before performing any formal numerical calculation, make a reasoned qualitative estimate: which outcome do you think is more likely, that all 4 households sort packaging ($X = 4$) or that exactly 2 of them do ($X = 2$)? Justify your conjecture based on the probability of success of each individual household.
2. b) Model the experiment using a binomial distribution, explicitly stating its parameters, and accurately calculate the probabilities $P(X = 4)$ and $P(X = 2)$.
3. c) Check whether the values obtained support your initial estimate and reason how the skewness of the distribution (due to $p > 0,5$) justifies the quantitative difference between both outcomes.

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").

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 sports centre, $75 \%$ of the users enrolled in swimming lessons are able to swim 100 metres 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, 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?

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).

A local digital commerce platform sends an exclusive discount code to a selection of 7 customers who have not made a purchase for a month. From past experience, it is known that the probability of any customer using the code before it expires is $p = 0,25$. The sales manager needs to know the probability that at least 2 of the 7 customers redeem the discount ($P(X \geq 2)$).

1. a) To solve the problem, two calculation approaches are proposed:

• Approach A (direct): Calculate $P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7)$.

• Approach B (complementary event): Calculate $1 - [P(X = 0) + P(X = 1)]$.

Argue which of the two approaches is preferable in practice to optimize calculation time and minimize the propagation of rounding errors.

1. b) Apply the most efficient approach to calculate the value of $P(X \geq 2)$, rounding the result to four decimal places.
2. c) Suppose that in a different campaign with another 7 customers, the redemption probability were $p = 0,90$ and one wished to calculate the probability that at least 6 of them make a purchase ($P(X \geq 6)$). Would the complementary event approach still be the fastest, or would the direct approach be better in that case? Justify your answer without needing to perform the numerical calculations.

According to data from the Active Population Survey (EPA) in Spain, in the advanced services sector approximately $20\,\%$ of employed individuals regularly work in teleworking mode ($p = 0,2$). A labor consultancy firm randomly selects a sample of 12 workers from this sector.

1. a) Justify in detail why the number of teleworkers in the sample can be modeled using a binomial random variable $X \equiv B(n, p)$, verifying the conditions of independence and constancy of probability.
2. b) Calculate the mean or expected value ($\mu$) and the standard deviation ($\sigma$) of the number of teleworkers in this sample. Interpret the practical meaning of both statistics in the company's context.
3. c) The consultancy firm considers that the sample "is not representative for a qualitative analysis" if it contains fewer than 2 teleworkers ($X < 2$). Calculate the probability that this unfavorable scenario occurs and interpret the feasibility of the study based on the obtained result.

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?

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?

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

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 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.

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.

In a nationwide express courier logistics company, service quality is audited daily by analyzing the number of packages that experience delivery issues or delays. This quantity follows a binomial model $X \equiv B(n, p)$.

Following an issue on the control server, the records containing the initial parameters were deleted from the system: the size of the daily audited batch ($n$) and the individual probability of delay ($p$). However, the consolidated statistical report accurately preserves two summary measures of the process:

• The mean number of daily shipments with an issue is $\mu = 10$.

• The variance of shipments with an issue is $\sigma^2 = 9$ (that is, a standard deviation $\sigma = 3$).

1. a) Set up the system of equations derived from the formulas for the mean and variance of a binomial distribution, and algebraically solve for the original values of parameters $n$ and $p$.
2. b) The operations director proposes doubling the size of the audit to $2n$ packages per day while maintaining the same quality standard $p$, claiming that with this measure "the dispersion or standard deviation of delays will also double." Prove analytically whether his statement is true or false by deriving the exact expression for the new standard deviation.
3. c) Using the original parameters obtained in part a), calculate the probability that on a given day no issues occur ($P(X = 0)$). Express the result in scientific notation and critically reflect on its meaning in the day-to-day operations of the company.

A public ecological transition agency is conducting an exploratory study on the use of photovoltaic solar panels for self-consumption in single-family homes in a county. A random sample of $n = 4$ independent homes is selected.

In the analysts' theoretical models, it is established that the probability that none of the 4 homes have solar panels ($P(X = 0)$) is exactly 16 times greater than the probability that all 4 homes have them ($P(X = 4)$).

1. a) From the condition $\frac{P(X = 0)}{P(X = 4)} = 16$, set up an equation relating $p$ and $q = 1 - p$. Solve it step by step to deduce the exact value of the individual probability $p$ that a randomly selected home has solar panels.
2. b) With the obtained value of $p$, calculate the probability that in the sample of 4 homes there is a strict majority with a photovoltaic installation (that is, $X \geq 3$).
3. c) Imagine that the 4 selected homes belong to the same private residential development in which the homeowners' association agreed at a meeting to install panels jointly to reduce costs through a collective subsidy. Reason critically whether the binomial model $B(4, p)$ would still be valid in this circumstance, or which fundamental assumption of the Bernoulli experiment would be invalidated.

Unit challenge. How do official statistics fit your environment? Prepare an executive report comparing digital habits with probabilistic models.

1. Real data: Check the INE website for the proportion of young people who shop online ($p \approx 0,65$). Then, survey $10$ acquaintances to record how many did so in the last month ($n = 10$).

2. Binomial distribution: Model the discrete variable $X \sim B(10; 0,65)$ in a spreadsheet or GeoGebra. Obtain the probability table $P(X = k)$, the distribution function $F(x)$, the mean $\mu$, and the standard deviation $\sigma$. Compare the expected value with your sample result.

3. Leap to the continuous variable: Also record the daily screen time hours of your respondents (continuous variable $Y$). Using the histogram, associate the data with a probability density function $f(y)$ and theoretically explain why, for the continuous variable, $P(Y = 3) = 0$, unlike in the binomial.

4. Final report: Integrate your conclusions, tables, and comparative graphs into a short document.

Assessment will consider:

• Rigour in calculating probabilities and parameters of the binomial distribution.

• Conceptual understanding when differentiating discrete distributions from continuous probability density and distribution functions.

• Rigorous data extraction from the INE and the quality of the graphs generated with GeoGebra or spreadsheet software.