Exercises: Contingency tables

In a study on social media use in a secondary school, 100 2nd year Bachillerato students were surveyed about their preferred platform. The data are shown in the following contingency table, although some values have been erased:

Given that there is a total of 45 boys and 55 girls:

1. a) Complete the missing values in the table.
2. b) If we choose a student at random, what is the probability that their preferred network is TikTok?
3. c) What is the probability that it is a girl who prefers Instagram?

A streaming platform analyses its users' preferences according to their age. A random sample is selected, and the following results are obtained:

Calculate the following probabilities when choosing a user at random:

1. a) $P(\text{Series})$, that is, the probability that they prefer series.
2. b) $P(\text{Films} \cap \text{Under 25})$.
3. c) $P(\text{Over 25} \cup \text{Films})$.

In a padel tournament organised by the local council, 80 people participate. It is known that 32 are men and the rest women. Of the men, 20 are federated, whereas among the women only 12 are federated.

1. a) Organise the information in a contingency table.
2. b) If we select a participant at random, calculate the probability that they are not federated.
3. c) What is the probability that the participant is female, given that she is federated?

A survey was conducted on transport habits for going to university. $40\%$ of the students are male ($H$) and $60\%$ are female ($M$). $25\%$ of the males go by car ($C$), whereas $15\%$ of the females use this mode of transport.

1. a) Considering a sample of 100 people, construct the contingency table with absolute values.
2. b) Calculate the probability that a randomly chosen person is female and does not use a car.
3. c) Calculate the probability that, given they are male, they do not use a car.

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

Observe the following table classifying a company's employees according to their department and whether they have requested remote working:

Calculate with reasoning using the table:

1. a) $P(T | V)$, the probability that an employee works remotely if they belong to the Sales department.
2. b) $P(I | T)$, the probability that an employee is from R&D given that they have requested remote working.
3. c) What is more likely: that a Sales employee works remotely or that an R&D employee works remotely?

In a language academy, students are divided by level (Basic or Advanced) and by language (English or German). The probability of a student being in the advanced level is $0,4$. The probability of studying English is $0,7$. It is known that the probability of a student studying German and being at the basic level is $0,2$.

1. a) Draw up a contingency table with probabilities (where the total is $1$).
2. b) Calculate the probability that a randomly chosen student is at the advanced level and studies English.
3. c) If a student studies English, what is the probability that they are at the basic level?

A group of 200 teenagers was asked whether they have cereals or fruit for breakfast. The results were: 120 have cereals for breakfast, 100 have fruit for breakfast and 40 have both.

1. a) Complete the following contingency table to organise the data:

1. b) What is the probability that a teenager has neither cereals nor fruit for breakfast?
2. c) Given that a teenager has cereals for breakfast, what is the probability that they also have fruit for breakfast?

A sports clothing shop classifies its trainers sales according to type (Running or Trail) and gender (Men or Women). On one day, they sold 50 pairs:

Calculate the probability that, choosing a sale at random:

1. a) The trainers are Trail.
2. b) They are for a woman, given that they are Running trainers.
3. c) They are Trail, given that it is a sale to a man.

In a road safety campaign at a high school, the mode of commuting of $120$ Bachillerato students is analysed. The data indicate that $40$ commute by bicycle (of which $28$ wear a helmet and $12$ do not) and the remaining $80$ commute on foot or by public transport (none of whom wear a helmet).\n\nTo calculate the probability that a student wears a helmet knowing that they commute by bicycle, Mateo reasons as follows:\n«Since there are $28$ cyclists with a helmet out of the $120$ surveyed students, the requested probability is $P(\text{Helmet}|\text{Bike}) = \frac{28}{120} \approx 0,233$, that is, barely $23,3\,\%$ of cyclists wear a helmet».\n\na) Identify Mateo's conceptual error and explain what «conditioning» means in a contingency table in terms of reducing the sample space.\nb) Correct the calculation and find the correct value of $P(\text{Helmet}|\text{Bike})$.\nc) Which probabilistic event does the fraction $\frac{28}{120}$ calculated by Mateo actually correspond to? Write its formal notation.

A mobility audit evaluates the punctuality of $200$ Cercanías Renfe journeys on two suburban lines, $C_1$ and $C_2$, obtaining the following information:\n- On line $C_1$, $120$ journeys were recorded, of which $18$ were delayed.\n- On line $C_2$, $80$ journeys were recorded, of which $20$ were delayed.\n\na) Before performing any formal numerical calculation, make a reasoned estimate at a glance: which of the two lines seems to you to have a higher percentage of delays? Justify your intuition by mentally comparing the proportion of delays relative to the total number of journeys on each line.\nb) Transfer all the data into a complete $2 \times 2$ contingency table (Lines $C_1$ and $C_2$ in rows; On time and Delayed in columns), calculating all marginal totals.\nc) Formally verify your initial estimate by calculating the conditional probabilities $P(\text{Delayed}|C_1)$ and $P(\text{Delayed}|C_2)$. Do the exact results confirm your qualitative prediction?

In a study on digital content consumption habits among $200$ teenagers, it has been observed that $120$ are subscribed to a streaming platform. Of those who are subscribed, $80$ prefer watching series over films. Among those who do not pay for a subscription, $30$ prefer films.

1. a) Organise the above information in a contingency table.
2. b) If we choose a teenager at random, what is the probability that they prefer watching series?
3. c) Given that the chosen teenager prefers films, what is the probability that they are not subscribed to any platform?

A technology company analyses the relationship between the use of a new model of wireless headphones and customer satisfaction. Of the $500$ users surveyed, $350$ bought the "Pro" model and the rest the "Basic" model. It is known that event $S$ (being satisfied) and event $M$ (Pro model) are independent. If the probability of being satisfied is $P(S) = 0,8$:

1. a) Complete the contingency table with the expected number of people for each category.
2. b) Calculate the probability that a user bought the "Basic" model and is not satisfied.
3. c) What does it mean in this context for the events to be independent?

In a secondary school, $60\%$ of the 2nd-year Bachillerato students are female. It is known that $20\%$ of the girls take part in the debate club, while $10\%$ of the boys also do.

1. a) Build a contingency table of relative frequencies (probabilities).

1. b) If we select a student at random, what is the probability that they belong to the debate club?

1. c) If the chosen student does not belong to the debate club, what is the probability that they are male?

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

A survey on voting intention and main social concern (Environment, Economy or Health) was conducted, obtaining the following partial probability table:

1. a) Complete the contingency table.
2. b) Calculate the probability that a Party A voter is concerned about Health.
3. c) Are the events "Voting for Party B" and "Being concerned about the Environment" independent? Justify your answer mathematically.

A sports centre offers Yoga, Pilates and Crossfit classes. The distribution of the $300$ members by activity and age group is as follows:

1. a) If we choose a member at random, what is the probability that they practice Yoga or are Young?
2. b) Among Adult members, which activity is the most probable?
3. c) If a Crossfit practitioner is chosen, what is the probability that they are Young?

The success of an advertising campaign on social media is analysed. Of the people who saw the ad, $40\%$ did so on Instagram and $60\%$ on TikTok. It is known that $15\%$ of those who saw it on Instagram bought the product, while on TikTok $10\%$ did.

1. a) Imagine a group of $1000$ people who have seen the ad and transfer these percentages to a contingency table (rows: buys / does not buy; columns: Instagram / TikTok, with their totals).

1. b) Calculate the probability that a randomly chosen person has bought the product.

1. c) Given that a person has bought the product, on which social network is it more likely that they saw the ad?

In a medical study on the effectiveness of a vaccine, the following data are obtained from a sample of $1000$ patients:

1. a) Complete the table.
2. b) Calculate the probability of becoming ill having been vaccinated.
3. c) Calculate the probability of becoming ill having not been vaccinated.
4. d) In view of the results, do you consider that the vaccine is effective? Justify your answer by comparing conditional probabilities.

An e-commerce platform classifies its customers into "Frequent" and "Occasional", and their purchases into "Electronics" or "Fashion". It is known that $30\%$ of customers are Frequent. Of all purchases made, $40\%$ are Electronics. Furthermore, $10\%$ of total customers are Frequent who bought Fashion.

1. a) Construct the complete contingency table of probabilities.
2. b) What is the probability that a frequent customer buys Electronics?
3. c) Given that a customer bought Electronics, what is the probability that they are an Occasional customer?

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

From a representative sample of $500$ young people aged between $20$ and $24$ years taken from the Labour Force Survey (EPA) in a region, the relationship between their educational attainment level and their current employment status is analysed:\n- $300$ young people have completed Vocational Training (FP), and of them, $240$ are employed.\n- In the total set of the $500$ surveyed young people, there are exactly $140$ unemployed people.\n- The rest of the young people in the sample have at most a Bachillerato certificate.\n\na) Model this situation by constructing the complete contingency table of absolute frequencies cross-referencing the two variables (Educational level vs. Employment status) with their corresponding marginal totals.\nb) If a young person from the sample is chosen at random, calculate the probability that they are unemployed knowing that they only have Bachillerato, $P(\text{Unemployed}|\text{Bachillerato})$, and compare it with $P(\text{Unemployed}|\text{FP})$.\nc) Mathematically justify whether, in this sample, being unemployed is an event independent of having studied Vocational Training. What socio-economic conclusion can be drawn from the result?

In a bank logistics sorting centre, $1000$ suspicious banknotes are examined using an automatic scanner. It is known that $40$ of them are counterfeit and $960$ are legal tender. The scanner triggers the alarm on $90\,\%$ of the counterfeit banknotes, but generates a false alarm on $5\,\%$ of the legitimate banknotes.\n\nWe want to calculate the probability that a banknote is truly counterfeit if the scanner has triggered the alarm: $P(\text{Counterfeit}|\text{Alarm})$.\n\na) Two students discuss how to solve it: Lucía suggests applying the theoretical formula of Bayes' Theorem using a probability tree, while David prefers to transfer the absolute frequencies into a $2 \times 2$ contingency table. Construct the contingency table with the values proposed by David.\nb) From the obtained table, directly calculate the conditional probability $P(\text{Counterfeit}|\text{Alarm})$.\nc) Choose and justify which of the two methods (tree with formulas or contingency table) best allows one to visualise the impact of false alarms on the total number of warnings. Explain why the obtained result usually seems surprisingly low at first glance.

A cybersecurity department analyses the email traffic of a large technology company. It is known that $12\%$ of the emails entering the server are phishing attempts (fraud). The smart filtering system labels emails as "Suspicious" or "Safe".

The system is able to correctly detect $95\%$ of phishing emails. However, due to overcautiousness, $3\%$ of legitimate emails are wrongly labelled as "Suspicious".

1. a) Organise the information in a contingency table using a theoretical sample of $10\,000$ emails.

1. b) If an employee receives an email labelled as "Suspicious", what is the probability that it is actually a legitimate email?

1. c) Are the events "Phishing Email" and "Labelled as Suspicious" independent events? Justify your answer by calculating probabilities.

In a study on the streaming content consumption habits of young people between 18 and 22 years old, the following data were obtained:

• $65\%$ of young people are subscribed to Netflix ($N$).

• $40\%$ are subscribed to Disney+ ($D$).

• The probability that a young person is not subscribed to either of the two platforms is $0,25$.

1. a) Draw up a contingency table expressing the values in terms of probability.

1. b) A young person is chosen at random and turns out to be subscribed to Netflix. Calculate the probability that they are also subscribed to Disney+.

1. c) Determine whether having a subscription to Netflix is independent of having a subscription to Disney+.

A second-hand clothing sales platform analyses the behaviour of its users. $60\%$ of the users are "Repeat Buyers" ($C$), while the rest are "Occasional Buyers".

It is observed that $70\%$ of repeat buyers use the mobile app for their transactions, whereas only $30\%$ of occasional buyers use it (preferring the website).

1. a) If we select a user at random, what is the probability that they use the website to make their purchases?

1. b) If we know that a user has made a purchase via the website, what is the probability that they are a repeat buyer?

1. c) Represent the situation using a contingency table for a population of $500$ users.

A vocational college offers two higher-level courses: Web Application Development ($DAW$) and Marketing and Advertising ($MK$). $55\%$ of the students are enrolled in $DAW$.

It has been found that $20\%$ of the students in $DAW$ and $15\%$ of those in $MK$ drop out of their studies before finishing the first year.

1. a) Construct the contingency table associated with this educational centre.

1. b) If we choose a student at random from among those who dropped out of the course, what is the probability that they were enrolled in Marketing and Advertising?

1. c) Is dropping out of studies independent of the course in which the student is enrolled? Reason your answer by comparing $P(Abandono)$ with $P(Abandono | DAW)$.

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

In a survey conducted at a music festival about musical genre preference and age, the following data were obtained from a sample of $200$ attendees:

Knowing that the total number of Pop lovers is $65$ and that attendees under 25 represent $45\%$ of the total sample:

1. a) Complete the contingency table with the absolute values.

1. b) Calculate the probability that a randomly chosen attendee is older than 25 and likes Indie.

1. c) If an attendee likes Rock, what is the probability that they are under 25 years old?

A mobile payment platform uses an automated fraud detection system. System data indicate that it correctly detects $98\%$ of actually fraudulent transactions ($F$). On the other hand, it generates a "false alarm" (labels a legitimate transaction, $L$, as suspicious) in $4\%$ of cases. On the platform, it is estimated that $5\%$ of transactions are fraudulent.

1. a) Create a contingency table for a sample of $10\,000$ transactions.

1. b) If a transaction has been flagged as suspicious, what is the actual probability that it is fraudulent? Comment on the result obtained in relation to the reliability of the system.

1. c) If the system does not flag a transaction, what is the probability that it is actually legitimate?

A gym offers two types of activities: Weight Training ($M$) and Group Classes ($C$). $40\%$ of the members are female. It is known that $70\%$ of female members choose group classes, while $80\%$ of male members prefer weight training.

1. a) If we choose a member at random, what is the probability that they prefer weight training?

1. b) If we select a member who prefers group classes, what is the probability that they are male?

1. c) Determine whether the member's sex and the preference for bodybuilding are independent events. Explain your mathematical reasoning.

In a study on the use of electric vehicles ($E$) and combustion vehicles ($C$) in a city, $1\,200$ drivers are analysed. $25\%$ of drivers have an electric vehicle. Of the electric vehicle drivers, $60\%$ live in detached houses ($U$), whereas among the combustion vehicle drivers, this percentage falls to $20\%$.

1. a) Calculate the probability that a randomly chosen driver lives in a detached house.

1. b) If a driver lives in a detached house, what is the probability that they own an electric vehicle?

1. c) Are the events "Having an electric vehicle" and "Living in a detached house" independent? Prove it by comparing the relevant probabilities.

In a clinical trial with $400$ patients suffering from chronic migraine, a new drug is evaluated against a placebo. The sample is divided into $240$ patients receiving the drug and $160$ receiving the placebo. At the end of the trial, it is found that exactly $240$ of the total $400$ patients experienced a noticeable improvement. We denote by $x$ the number of patients in the placebo group who experienced improvement.\n\na) Model the contingency table in terms of the parameter $x$ and determine what exact value $x$ must take for the patient's improvement to be statistically independent of the treatment received.\nb) Suppose that, after unblinding the trial protocol, the actual data reveal that $x = 40$. Calculate the conditional probabilities of improvement in each group, $P(\text{Improvement}|\text{Drug})$ and $P(\text{Improvement}|\text{Placebo})$, and demonstrate whether or not the drug is effective compared to the placebo.\nc) Inverse problem: Keeping the marginal totals of the table fixed ($240$ with drug, $160$ with placebo, and $240$ improved in total), determine the complete interval of integer values that $x$ can take so that we can state with statistical rigour that the drug is strictly more effective than the placebo. Justify the boundary values by analysing the natural non-negativity constraints of the cells.

In a championship with $2000$ federated athletes, the anti-doping committee estimates that $2\,\%$ consume a prohibited substance. To detect it, a test is used with a sensitivity of $95\,\%$ (probability of testing positive if the athlete is doping) and a false positive rate of $5\,\%$ (probability of testing positive if the athlete is clean).\n\nAn athlete tests positive in the control. The following day, a sports newspaper publishes the headline: 'The test has a $95\,\%$ reliability, so the probability that the sanctioned athlete is guilty is $95\,\%$. The defence lawyer, on the other hand, claims before the court that 'with these same data, the probability that their client is innocent despite the positive test exceeds $70\,\%$.'\n\na) Construct the complete contingency table of absolute frequencies for the $2000$ athletes (Doping / Clean versus Test Positive / Test Negative).\nb) Identify the probabilistic error made by the newspaper (known as the fallacy of the transposed conditional). Which two conditional probabilities has the journalist confused?\nc) Using the table, calculate $P(\text{Doping}|\text{Positive})$ and $P(\text{Clean}|\text{Positive})$. Determine with mathematical rigour who was right (the journalist or the lawyer) and reflect critically on the ethical risk of applying diagnostic tests with false positives when the prevalence of the event in the population is very low.