Exercises: Tree diagrams and contingency tables

In a study on social media use in a school in Mexico City, it was observed that in a group of $100$ high school students, $65$ use TikTok regularly and $45$ use Instagram. Furthermore, it is known that $25$ students use both platforms.

Represent this data in a contingency table and calculate the probability that, when choosing a student at random:

1. a) They do not use either of the two social networks.
2. b) They use TikTok, given that they already use Instagram.

A box of tech supplies contains $8$ white wireless mice and $4$ black ones. If two mice are drawn sequentially without replacement:

1. a) Draw the tree diagram describing all possible outcomes, indicating the probabilities on each branch.
2. b) Calculate the probability that both mice drawn are the same color.

A 2nd Year Baccalaureate student sits a final Physics exam. It is known that $70\%$ of the students who studied using the school's digital platform passed, whereas only $40\%$ of those who did not use it passed. It is estimated that $60\%$ of the class used the platform.

Using a tree diagram, determine:

1. a) The probability that a student chosen at random passed the exam.
2. b) Given that a student failed, what is the probability that they did not use the digital platform?

In a language academy, students are divided according to the course they take: $50\%$ study English, $30\%$ French and $20\%$ German. In addition, it is known that the percentage of women in each course is $60\%$, $55\%$ and $45\%$ respectively.

If a student is chosen at random, construct a contingency table (considering a hypothetical total of 1 000 students) and calculate:

1. a) The probability that the student is male.
2. b) Given that the student is male, the probability that he studies English.

We have two urns prepared for a draw. Urn A contains $3$ green balls and $2$ red ones. Urn B contains $1$ green ball and $4$ red ones. The experiment consists of flipping a coin: if it comes up heads, we draw a ball from Urn A; if it comes up tails, we draw it from Urn B.

1. a) Represent the situation using a tree diagram.
2. b) Calculate the probability that the ball drawn is red.

A cell phone repair shop receives devices with two main types of fault: broken screen or defective battery. Last month, out of the $200$ cell phones repaired, $120$ had a broken screen, $70$ had battery problems and $30$ had both faults.

Organize the information in a contingency table and answer:

1. a) What is the probability that a cell phone chosen at random has only a broken screen?
2. b) If the cell phone has a defective battery, what is the probability that it also has a broken screen?

In an electronic component manufacturing company, two machines, $M_1$ and $M_2$, are used to produce chips. Machine $M_1$ produces $60\%$ of the chips and the rest are produced by $M_2$. $2\%$ of the chips from $M_1$ are defective, whereas for $M_2$ the percentage of defective chips is $5\%$.

1. a) Construct the tree diagram associated with this production process.
2. b) If a chip is selected at random and turns out to be defective, what is the probability that it was manufactured by machine $M_1$?

A survey was conducted with $80$ people about their reading habits in digital or paper format. The results show that $32$ people read exclusively on paper, $28$ people read in both formats and the rest read exclusively in digital format.

1. a) Draw up a contingency table summarising these data.
2. b) Calculate the probability that a person chosen at random reads in digital format.
3. c) Given that the person reads on paper, what is the probability that they also read in digital format?

A streaming platform offers two types of subscriptions: Standard and Premium. It is known that $60 \%$ of users have the Standard subscription, while the rest have the Premium one. Of the users with a Standard subscription, $25 \%$ watch content in 4K resolution, while among Premium users, this percentage rises to $80 \%$.

Represent the situation using a tree diagram and calculate:

1. a) The probability that a randomly chosen user watches content in 4K.

1. b) If we know that a user is watching a movie in 4K, what is the probability that they have a Standard subscription?

A survey on social media usage across Bachillerato grades was conducted in a high school in Madrid. The data have been arranged in the following contingency table, although some values are missing:

	Instagram ($I$)	TikTok ($T$)	Total
1st year Bachillerato	$45$		$110$
2nd year Bachillerato		$60$
Total	$85$		$210$

1. a) Complete the contingency table in your notebook.
2. b) If a student is chosen at random and turns out to be in 2nd year of Bachillerato, what is the probability that their preferred social network is Instagram?
3. c) Are the events "being in 1st year of Bachillerato" and "preferring TikTok" independent? Justify your answer mathematically.

A technology company manufactures drone components in three different factories: $F_1$, $F_2$ and $F_3$. $40 \%$ of total production comes from $F_1$, $35 \%$ from $F_2$ and the rest from $F_3$. Quality control indicates that the percentage of defective components is $3 \%$ in $F_1$, $2 \%$ in $F_2$ and $5 \%$ in $F_3$.

1. a) Calculate the probability that a component chosen at random from the overall production is not defective.
2. b) If a customer buys a component and it turns out to be defective, in which factory is it most likely to have been manufactured? Justify your result using Bayes' Theorem.

A data analyst studies the relationship between public transport use and employee punctuality in a large office. After observing $500$ workers, they find that $350$ use public transport and the rest travel by private vehicle. Of those using public transport, $10 \%$ arrive late, whereas of those travelling by private vehicle, $20 \%$ arrive late.

1. a) Organise this information into a contingency table of absolute frequencies.
2. b) What is the probability that a worker chosen at random arrives on time?
3. c) If a worker has arrived late, what is the probability that they came by public transport?

In an esports tournament, the organising committee performs performance tests on the players. It is estimated that $5 \%$ of participants use some kind of prohibited software (cheating). The test used to detect these cheats has a reliability of $98 \%$ (detects the software if the player is cheating), but has a $2 \%$ rate of "false positives" (indicates cheating when the player is legitimate).

1. a) If a player is chosen at random, what is the probability that the test gives a positive result (indicating cheating)?
2. b) If a player's test has given a positive result, what is the actual probability that they are cheating? Interpret the result obtained.

In a vocational course, $10$ students are undertaking work placements, of whom $6$ are at tech companies and $4$ at industrial companies. Two students are chosen at random, consecutively and without replacement.

1. a) Draw the tree diagram showing the probabilities on each branch.
2. b) Calculate the probability that both chosen students are at the same type of company.
3. c) Calculate the probability that at least one is at a tech company.

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

The reading habits of two groups of people are studied: Young people (under 25 years old) and Adults. It is known that $70 \%$ of respondents are Adults. $40 \%$ of Young people read at least one book a month, whereas $60 \%$ of Adults do so.

1. a) Construct the contingency table of associated probabilities (relative frequencies).
2. b) Calculate the probability that a person chosen at random reads at least one book a month.
3. c) If a person states that they do not read any books a month, what is the probability that they are Young?

A GPS navigation app estimates arrival time at a destination based on traffic. If traffic is light (probability $0.7$), the app estimates the arrival time correctly $90 \%$ of the time. If traffic is heavy, it only gets it right $40 \%$ of the time.

1. a) Calculate the probability that the app is incorrect in its time estimation.
2. b) If today the app failed in its prediction, what is the probability that the traffic was light?

An IT security system for critical infrastructure in Mexico consists of three sequential filtering barriers to detect intrusion attempts (brute-force attacks).

1. Barrier A detects $95 \%$ of the attacks, but lets the rest pass through.

2. Barrier B receives what Barrier A has not detected. This barrier has a success rate of $80 \%$.

3. Barrier C is the last resort and detects $60 \%$ of the remaining attacks.

However, the system is not perfect and sometimes generates false alarms (detects legitimate traffic as an attack). It is known that the probability of a data packet being a real attack is $0.001$. The false alarm probabilities are $1 \%$ for Barrier A, $2 \%$ for Barrier B, and $5 \%$ for Barrier C.

Using a tree diagram, calculate:

1. a) The probability that a data packet is blocked by the system (that is, that it is detected by any of the three barriers).

1. b) If the system has just blocked a data packet, what is the probability that it is actually a cyberattack and not a false alarm?

In a prestigious esports academy, players are classified according to their performance into two categories: Elite ($E$) and Promising ($P$). A study has been conducted on the use of mechanical keyboards ($M$) and membrane keyboards ($\bar{M}$).

It is known that $30 \%$ of the players belong to the Elite category. Furthermore, among the Promising players, the probability of them using a membrane keyboard is twice that of them using a mechanical keyboard. Finally, it is known that the probability of a player chosen at random being Elite and using a mechanical keyboard is $0.24$.

1. a) Construct the complete contingency table with the probabilities of each event.

1. b) Calculate the probability that a player uses a mechanical keyboard.

1. c) If we select a player who uses a mechanical keyboard, what is the probability that they belong to the Promising category?

1. d) Are the events "being an Elite player" and "using a mechanical keyboard" independent? Justify your answer mathematically.

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

A pharmaceutical laboratory is analysing the efficacy of a new detection test for a respiratory virus variant. It is known that in the current population, the prevalence of the disease is $5 \%$.

The test has a specific property: if applied to a sick person, it gives a positive result in $98 \%$ of cases. If applied to a healthy person, it gives a negative result in $96 \%$ of cases. Suspecting an error in a mass testing campaign, it is decided that all people who test positive in the first test must undergo a second test that is identical to and independent of the first.

1. a) Represent the situation using a tree diagram for a person chosen at random from the population who undergoes the full process if necessary.

1. b) What is the probability that a person from the population tests positive in the second test (given that they already tested positive in the first)?

1. c) If a person has tested positive in both tests, what is the probability that they are actually sick? Comment on the result by comparing it with the probability after a single positive test.

In a technology workshop, two urns contain parts made of different materials for a project.

• Urn A contains 6 bioplastic (PLA) parts and 4 resin parts.

• Urn B contains 3 PLA parts and 7 resin parts.

The experiment is as follows: a part is drawn at random from Urn A and placed into Urn B without looking at its material. Then, two parts are drawn simultaneously from Urn B.

1. a) Calculate the probability that both parts drawn from Urn B are made of resin.

1. b) Given that both parts drawn from Urn B were made of resin, what is the probability that the part transferred from Urn A was PLA?

A recommendation algorithm of a music streaming platform uses three different models ($M_1, M_2, M_3$) to suggest songs to users. $40 \%$ of the time it uses $M_1$, $35 \%$ it uses $M_2$ and the rest of the time it uses $M_3$.

The probability that a user "likes" a song suggested by $M_1$ is $0.3$. If the song is suggested by $M_2$, the probability is $0.45$. It is known that the overall probability of a user "liking" a randomly chosen song is $0.4325$.

1. a) Calculate the probability that a user "likes" a song suggested by the third model, $M_3$.

1. b) If a user has "disliked" (the complementary event to "like") a song, what is the probability that it was suggested by model $M_1$?

1. c) Three suggested songs are selected independently. What is the probability that at least one of them receives a "like"?

In a survey conducted among 2nd Year Baccalaureate students about their study preferences after secondary school, the following data was obtained, partially represented in this contingency table. The variables are Gender ($H$ for male, $M$ for female) and Area of Interest ($I$ for Engineering, $S$ for Health, $O$ for Others).

Given that the events "being female" and "being interested in engineering" are independent events, and that there is the same total number of males and females in the sample of 200 people:

	Engineering ($I$)	Health ($S$)	Others ($O$)	Total
Male ($H$)	40
Female ($M$)		45
Total		75		200

1. a) Complete the contingency table, justifying your steps.

1. b) Calculate the probability that a student chosen at random is interested in Health or is male.

1. c) If we choose a student who is not interested in Engineering, what is the probability that they are female?

An autonomous exploration robot on Mars has two sensors to detect the presence of ice deposits beneath the surface. The Main Sensor detects ice with a probability of $0.9$ when it exists, but also gives a false positive (indicates ice when there is none) $5 \%$ of the time.

It is known that the robot is exploring an area where the prior probability of finding ice is $15 \%$.

The robot is programmed to send a confirmation signal only if the Main Sensor detects ice. However, to save energy, if the Main Sensor detects ice, a Backup Sensor is automatically activated, which has an accuracy of $95 \%$ (both for confirming presence and for confirming absence).

1. a) Calculate the probability that the robot sends a signal indicating that the Main Sensor has detected ice.

1. b) If the Main Sensor has detected ice, what is the probability that the Backup Sensor does not detect ice?

1. c) What is the probability that there is actually ice if both sensors have given a positive result?

Two friends, Lucía and Carlos, play a turn-based strategy video game. Lucía is more experienced and wins $60 \%$ of individual games (this is the probability that she wins the first game). They decide to play a best-of-three tournament (the first to win two games wins the tournament).

Due to fatigue and learning during play, the probabilities are not constant:

• If Lucía wins a game, the probability that she wins the next one increases to $0.7$.

• If Lucía loses a game, the probability that she wins the next one drops to $0.5$.

1. a) Draw the tree diagram that represents the course of the tournament, indicating the probability on each branch and, at the end of each path, who wins the tournament.

1. b) Calculate the probability that Carlos wins the tournament.

1. c) If the tournament lasted exactly three games, what is the probability that Lucía won it?

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