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:

	Films	Series	Total
Under 25 years old	$0.15$	$0.35$	$0.50$
Over 25 years old	$0.30$	$0.20$	$0.50$
Total	$0.45$	$0.55$	$1.00$

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 organized by the local municipality, 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) Organize 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 store classifies its sneaker sales according to type (Running or Trail) and gender (Men or Women). On one day, they sold 50 pairs:

	Running	Trail	Total
Men	18	12	30
Women	15	5	20
Total	33	17	50

Calculate the probability that, choosing a sale at random:

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

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:

Concern	Party A	Party B	Totals
Environment	$0.15$	$0.10$
Economy	$0.25$		$0.45$
Health		$0.12$
Totals			$1.00$

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

Activity	Young people (16-30)	Adults (>30)
Yoga	$20$	$50$
Pilates	$30$	$60$
Crossfit	$110$	$30$

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:

	Became ill	Did not become ill	Totals
Vaccinated	$20$		$600$
Unvaccinated		$280$
Totals			$1000$

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.

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 analyzes the behavior 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 center.

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? Justify your answer by comparing $P(Abandono)$ with $P(Abandono | DAW)$.

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:

	Pop ($P$)	Rock ($R$)	Indie ($I$)
Under 25 years old ($M$)	$40$		$30$
25 years old or older ($A$)		$50$

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.