Two-Way (Contingency) Tables

In a study on the use of social media in a high school, 100 12th-grade students have been surveyed about their preferred platform. The data are collected 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 group of 200 teenagers was asked whether they have cereal or fruit for breakfast. The results were: 120 have cereal, 100 have fruit, and 40 have both.

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

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

In a pickleball tournament organized by the local recreation department, 80 people participate. It is known that 32 are men and the rest are women. Of the men, 20 are registered, while of the women only 12 are registered.

1. a) Organize the information into a contingency table.
2. b) If we select a participant at random, calculate the probability that they are not registered.
3. c) What is the probability that the participant is a woman given that she is registered?

A survey has been carried out on transportation habits for going to college. $40\%$ of the students are men ($H$) and $60\%$ are women ($M$). $25\%$ of the men commute by car ($C$), while $15\%$ of the women use this mode of transportation.

1. a) If we consider a sample of 100 people, construct the contingency table with the absolute values.
2. b) Calculate the probability that a person chosen at random is a woman and does not use a car.
3. c) Calculate the probability that, being a man, he does not use a car.

The success of an advertising campaign on social media is analyzed. 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 so.

1. a) Transfer these percentages to a contingency table of $1000$ people.
2. b) Calculate the probability that a person chosen at random has bought the product.
3. c) Given that a person has bought the product, on which social network is it more probable that they saw the ad?

In a high school, $60\%$ of the seniors are women. It is known that $20\%$ of the women participate in the debate club, while $10\%$ of the men also do so.

1. a) Construct a contingency table of relative frequencies (probabilities).
2. b) If we select a student at random, what is the probability that they belong to the debate club?
3. c) If the chosen student does not belong to the debate club, what is the probability that they are a man?

A survey has been carried out on voting intention and the main social concern (Environment, Economy or Health), obtaining the following partial table of probabilities:

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

In a language school, students are divided by level (Basic or Advanced) and by language (English or German). The probability that a student is at the advanced level is $0,4$. The probability that they study English is $0,7$. It is known that the probability that a student studies German and is at the basic level is $0,2$.

1. a) Create a contingency table with probabilities (where the total is $1$).
2. b) Calculate the probability that a student chosen at random 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?

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

A streaming platform analyzes the preferences of its users 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) $P(\text{Movies} \cap \text{Under 25 years old})$.
3. c) $P(\text{Over 25 years old} \cup \text{Movies})$.

Observe the following table which classifies the employees of a company according to their department and whether they have requested to work from home:

Calculate with reasoning using the table:

1. a) $P(T | V)$, the probability that an employee works from home 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 to work from home.
3. c) Which is more likely: that a Sales employee works from home or that an R&D employee works from home?

A sports center offers Yoga, Pilates and Crossfit classes. The distribution of the $300$ members according to 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?

A sporting goods store classifies its sneaker sales according to type (Running or Trail) and gender (Men's or Women's). In one day they sold 50 pairs:

Calculate the probability that when choosing a sale at random:

1. a) The sneakers are Trail.
2. b) They are women's given that they are Running.
3. c) They are Trail given that it is a men's sale.

In a study on the 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 shows over movies. Among those who do not pay for a subscription, $30$ prefer movies.

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

In a medical study on the efficacy 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 not having been vaccinated.
4. d) In light of the results, do you consider the vaccine to be effective? Justify your answer by comparing the conditional probabilities.

A technology company analyzes the relationship between the use of a new model of wireless earphones 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 has 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 survey carried out at a music festival about musical genre preference and age, the following data has been obtained from a sample of $200$ attendees:

Given that the total number of Pop lovers is $65$, that those under 25 years old represent $45\%$ of the total sample, and that the events "Being under 25 years old" and "Indie" are independent:

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

1. b) Calculate the probability that an attendee chosen at random is over 25 years old and likes Indie.

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

In a study on the streaming content consumption habits of young people aged between 18 and 22, the following data has been obtained:

• $65\%$ of the 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) Create 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 cybersecurity department analyzes the email traffic of a large technology company. It is known that $12\%$ of the emails entering the server are phishing (fraud) attempts. The smart filtering system labels the emails as "Suspicious" or "Safe".

The system is capable of correctly detecting $95\%$ of the phishing emails. However, due to overzealousness, $3\%$ of legitimate emails are incorrectly labeled as "Suspicious".

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

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

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

A new antigen test is being tested for a recent variant of a virus. Clinical data indicates that the test is positive in $98\%$ of people who are actually infected ($I$). On the other hand, the test gives a "false positive" (positive in healthy people, $S$) in $4\%$ of cases. In the current population, it is estimated that $5\%$ of people have the virus.

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

1. b) If a person takes the test and tests positive, what is the actual probability that they are infected? Comment on the result obtained in relation to the reliability of the test.

1. c) If the test is negative, what is the probability that the person is actually healthy?

A community college offers two vocational programs: 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 institution.

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

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

A secondhand clothing sales platform analyzes the behavior of its users. $60\%$ of the users are "Recurring Buyers" ($C$), while the rest are "Sporadic Buyers".

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

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

1. b) If we know that a user has made a purchase through the web, what is the probability that they are a recurring buyer?

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

A gym offers two types of activities: Weightlifting ($M$) and Group Classes ($C$). $40\%$ of the members are women. It is known that $70\%$ of women opt for group classes, while $80\%$ of men prefer weightlifting.

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

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

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

In a study on the use of electric vehicles ($E$) and gas-powered vehicles ($C$) in a city, $1\,200$ drivers are analyzed. $25\%$ of the drivers have an electric vehicle. Of the electric vehicle drivers, $60\%$ live in single-family homes ($U$), whereas among gas-powered vehicle drivers, this percentage drops to $20\%$.

1. a) Calculate the probability that a driver chosen at random lives in a single-family home.

1. b) If a driver lives in a single-family home, what is the probability that they own an electric vehicle?

1. c) Are the events "Owning an electric vehicle" and "Living in a single-family home" independent? Prove this by comparing the relevant probabilities.