Exercises: Tree diagrams and contingency tables

In a study on the use of social media at a secondary school in Dublin, it has been observed that, in a group of $100$ Leaving Certificate students, $65$ regularly use TikTok and $45$ use Instagram. In addition, it is known that $25$ students use both platforms.

Represent this data in a contingency table (with the rows "TikTok" and "No TikTok", the columns "Instagram" and "No Instagram", and the row and column totals) and calculate the probability that, when choosing a student at random:

1. a) They use neither of the two social networks.

1. 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 colour.

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

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 mobile phone repair shop receives devices with two main types of faults: broken screen or defective battery. Last month, of the $200$ 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 randomly chosen phone has only a broken screen?

1. b) If the 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?

Seán and Aoife are preparing an end-of-year charity raffle with an urn containing exactly $10$ seemingly identical USB flash drives: $3$ are $64\text{ GB}$ and $7$ are $128\text{ GB}$. The game consists of drawing two flash drives consecutively without replacement. Seán wants to calculate the probability that both flash drives are $64\text{ GB}$ and writes on the board:

$$P(\text{both are } 64\text{ GB}) = \frac{3}{10} \times \frac{3}{10} = \frac{9}{100} = 0.09$$

When Aoife warns him that the result is wrong, Seán defends himself by saying: "The proportion of $64\text{ GB}$ flash drives in the box is $3$ out of $10$; that probability is a fixed property of the flash drives and cannot change simply because I take one out."

1. a) Explain Seán's conceptual error with reasoning. What rule of tree diagrams has he violated, and what does it mean in this context that the branches of the second stage have conditional probabilities written on them based on what has already occurred?
2. b) Model the experiment by describing the branches that would emerge from each node in a correct tree diagram (with their precise probabilities).
3. c) Calculate the actual probability that both flash drives are $64\text{ GB}$ and the probability that at least one of the two is $128\text{ GB}$.

In a secondary school in Galway, $200$ students are enrolled in 6th Year (Leaving Certificate). Of these, $120$ are taking the Science and Technology track, while the remaining $80$ are taking Humanities and Social Sciences. Among the Science students, $30$ have chosen French as an elective. Meanwhile, among the Humanities and Social Sciences students, $40$ study French.

1. a) Two ways of organizing the information are proposed: a sequential tree diagram or a double-entry contingency table ($2 \times 2$). Explain why one of the two representations is more direct, clean, and appropriate when the data describe simultaneous characteristics of a fixed population rather than successive temporal stages.
2. b) Construct a contingency table by completing all its interior cells (intersections) and its marginal totals by rows and columns based on the provided data.
3. c) Using exclusively the sums and cells of the table, calculate the probability that a randomly chosen student takes the Humanities and Social Sciences track, given that they do not study French.

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 Senior Cycle year groups was conducted in a secondary school in Galway. The data have been arranged in the following contingency table, although some values are missing:

	Instagram ($I$)	TikTok ($T$)	Total
5th Year	$45$		$110$
6th Year		$60$
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 6th Year, what is the probability that their preferred social network is Instagram?
3. c) Are the events "being in 5th Year" 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?

On the Dublin-Cork intercity rail corridor, three passenger transport operators compete: Irish Rail, which operates $50\,\%$ of the daily journeys; Shannon Express, which operates $30\,\%$; and Emerald Rail, which covers the remaining $20\,\%$. According to punctuality data, the probability that a train suffers a delay of more than $15$ minutes is $3\,\%$ for Irish Rail, $5\,\%$ for Shannon Express, and $10\,\%$ for Emerald Rail.

1. a) Model the choice and journey of a randomly chosen passenger using a tree diagram showing all simple events, their marginal probabilities, and the conditional probabilities of delay.
2. b) If a randomly selected train reaches its destination without delay (that is, on time or with less than $15$ minutes of delay), what is the probability that the service was operated by Emerald Rail?
3. c) A consumer association argues on social media: "Since Emerald Rail has by far the highest delay rate ($10\,\%$), when a train arrives delayed at Heuston station, it is most likely to belong to Emerald Rail." Formally demonstrate, by calculating the posterior probabilities of each company for a delayed train, whether this statement is true or false, and explain what weight market share carries in the conclusion.

An IT security system for critical infrastructure in Ireland 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, Aoife and Seán, play a turn-based strategy video game. Aoife 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 Aoife wins a game, the probability that she wins the next one increases to $0.7$.

• If Aoife 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 Seán wins the tournament.

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

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

In a criminal trial held in a Provincial Court, a biological sample found at the crime scene is analyzed. The expert report determines that the genetic profile found matches that of the defendant. The forensic laboratory certifies that the test has a sensitivity of $100\,\%$ (if a person is guilty, the test always matches) and that the probability of an innocent person presenting this same profile by pure genetic chance is $1$ in $10,000$ ($P(\text{Match} \mid \text{Innocent}) = 0.0001$).

During his closing argument, the prosecutor states: "Your Honors, the probability that an innocent person has this genetic profile is barely $0.01\,\%$. Consequently, the probability that the defendant is innocent is only $0.01\,\%$, which proves his guilt beyond any reasonable doubt with a probability of $99.99\,\%$."

To test this assertion, the court defines a reference population of $100,000$ people, among whom it is certain that the sole perpetrator of the crime is found.

1. a) Identify with mathematical precision the error in reasoning committed by the prosecutor (known as the "prosecutor's fallacy"). Write, using the formal notation of events and conditional probability, what probability the forensic report actually provides and what probability the prosecutor claims to have proven.
2. b) Model the situation of the $100,000$ suspects using a $2 \times 2$ contingency table (Guilty / Innocent versus DNA Match / No match), deducing the expected number of individuals occupying each of the four interior cells.
3. c) From the values in the contingency table, calculate the actual probability that the defendant is guilty given that his DNA matches the sample, that is, $P(\text{Guilty} \mid \text{Match})$.
4. d) Interpret the radical difference between the percentage claimed by the prosecutor and the value obtained in the previous section. Explain how this phenomenon relates to the size of the suspect pool and why a very reliable technical match does not automatically equate to guilt.