Exercises: Tree diagrams

On a shelf in a teenager's bedroom there are two shoeboxes. Box A contains $3$ pairs of white sneakers and $2$ pairs of black sneakers. Box B contains $2$ pairs of white sneakers and $4$ pairs of black sneakers. A student chooses a box at random (with equal probability) and, without looking, takes out a pair of sneakers.

1. a) Draw the tree diagram that represents this situation, indicating the probabilities on each branch.

1. b) Calculate the probability that the chosen sneakers are white.

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

A TikTok content creator observes that $70 \%$ of their videos are comedy videos and $30 \%$ are challenge videos. The probability of a comedy video going viral is $0,1$, whereas for challenge videos that probability rises to $0,4$.

1. a) Represent the situation using a tree diagram.

1. b) If we choose a video at random from their profile, what is the probability that it is a viral video?

In an e-sports tournament, a player has to pass two stages. In Stage 1, the probability of success is $0,8$. If they pass the first, they move on to Stage 2, where the probability of success is $0,6$. If they do not pass the first stage, they are eliminated directly.

1. a) Draw the corresponding tree diagram.

1. b) Calculate the probability that the player successfully completes both stages of the tournament.

A 2nd-year Bachillerato student has prepared $60 \%$ of the topics for a Geography exam. The exam consists of drawing a topic at random from a bag. If the topic drawn is one of those he has studied, the probability of passing is $0,9$. If the topic is one of those he has not studied, the probability of passing drops to $0,2$.

1. a) Draw a tree diagram with the data from the problem, indicating the probability on each branch.

1. b) What is the total probability that the student passes the exam?

On a music streaming platform, $40 \%$ of users have a "Premium" account and $60 \%$ have a "Free" account. It is known that $80 \%$ of Premium users listen to music daily, while only $30 \%$ of free account users do so. A user is chosen at random:

1. a) Calculate the probability that they listen to music daily.

1. b) Given that the chosen user is Premium, what is the probability that they do not listen to music daily?

To get to school, a teenager takes the bus on $80 \%$ of days and walks the rest of the time. When taking the bus, they arrive late $5 \%$ of the time. However, when walking, they arrive late $15 \%$ of the time because they get distracted along the way.

1. a) Construct the tree diagram.

1. b) Calculate the probability that on any given day they arrive on time at school.

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

An urn contains $4$ red balls and $6$ blue balls. We draw two balls successively without replacement (without returning the first ball to the urn).

1. a) Represent the process using a tree diagram showing the probabilities on each branch. Take into account that the contents of the urn change after the first draw.

1. b) What is the probability that the two drawn balls are the same colour?

A sportswear shop receives sweatshirts from two factories. Factory 1 supplies $60 \%$ of the stock and Factory 2 the remaining $40 \%$. $2 \%$ of the sweatshirts from Factory 1 have some stitching defect, whereas in Factory 2 the percentage of defects is $5 \%$.

1. a) Represent the situation with a probability tree (draw it yourself: first the factory of origin and then whether or not the sweatshirt has a defect, indicating the probability of each branch).

1. b) Calculate the probability that a randomly chosen sweatshirt has no defect at all.

In an oil mill in Jaén, received olives are classified according to the harvesting method: traditional pole beating ($40\,\%$) or mechanical vibration ($60\,\%$). Due to bruising during harvesting, $10\,\%$ of the olives gathered by pole beating lose the optimal quality to produce "Extra Virgin" oil, while in mechanical harvesting that percentage rises to $25\,\%$. We want to calculate the probability that a random olive sample is suitable for "Extra Virgin" oil.

To solve this problem, two classmates propose different strategies:

• Sergio states: "It's more convenient to make up a total of $1.000$ kg of olives and build a contingency table with the kilos of each type".

• Alba argues: "It's much more direct and rigorous to set up a two-stage tree diagram, because the data given in the statement are already sequential conditional probabilities".

1. a) Compare both approaches: which of the two methods represents the structure of the information given in the statement more naturally, and why?
2. b) Solve the problem using the tree diagram: calculate the probability that a randomly selected sample yields "Extra Virgin" oil.
3. c) Justify whether the table method proposed by Sergio would have arrived at the exact same numerical result, and reflect on what methodological drawback inventing a hypothetical total of individuals might have compared to using a tree.

Lucía is solving a problem about practical driving tests at a driving school. $60\,\%$ of candidates pass on their first attempt. Of those who fail the first time, $70\,\%$ manage to pass on their second attempt. Lucía wants to calculate the probability that a randomly chosen person passes the driving test in one of the first two attempts.

In her notebook, she writes the following:

"I construct the tree: in the first stage, I have the branches Pass ($0,6$) and Fail ($0,4$). From the Fail branch, I draw two new branches: Pass on 2nd ($0,7$) and Fail on 2nd ($0,3$). Therefore, the probability of passing in one of the two attempts is the sum of passing on the first or on the second: $0,6 + 0,7 = 1,3$. Since it gives me more than $1$, the problem statement must be poorly written because a probability cannot be greater than $1$".

1. a) Identify the conceptual error that Lucía makes when operating with the branches of the tree and explain why her reasoning of directly adding $0,6$ and $0,7$ is incorrect.
2. b) Draw the correct tree diagram detailing the events at each level and the probabilities (marginal and conditional) of each branch.
3. c) Correct the calculation and find the actual probability that a candidate passes the driving test in one of the first two attempts.

A popular video game streamer broadcasts content on three different platforms: Twitch, YouTube and TikTok Live. According to his statistics, $45\,\%$ of his followers watch his live streams on Twitch, $35\,\%$ on YouTube and the remaining $20\,\%$ on TikTok. It is known that the percentage of followers who interact in the chat is $12\,\%$ on Twitch, $8\,\%$ on YouTube and $25\,\%$ on TikTok.

If we choose a follower of this streamer at random:

1. a) Represent the situation using a tree diagram.

1. b) Calculate the probability that the chosen follower interacts in the chat.

1. c) If we know that the chosen follower does not interact in the chat, what is the probability that they are a YouTube follower?

A health app uses a smart sensor to detect whether a user is performing intense aerobic exercise. The sensor is able to correctly detect the activity $98\,\%$ of the time. However, when the user is at rest or walking slowly, the sensor gives a "false positive" (erroneously indicating that they are doing intense exercise) in $3\,\%$ of cases. According to user community data, it is estimated that users spend $15\,\%$ of their time performing intense activity.

1. a) What is the probability that, at a moment chosen at random, the app indicates that the user is performing intense exercise?

1. b) If the app indicates that the user is performing intense exercise, calculate the probability that they are actually at rest or walking slowly.

In an E-sports tournament, a player must choose between two characters: "Warrior" or "Mage". The probability of choosing the Warrior is $60\,\%$. If he chooses the Warrior, he has a probability of winning the match of $0,7$. If he chooses the Mage, his probability of winning rises to $0,85$.

1. a) Draw the tree diagram associated with the experiment, indicating the probability on each branch.

1. b) Calculate the probability that the player wins the match.

1. c) If the player has lost the match, what is the probability that he had chosen the Warrior?

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

An online clothing store sends its orders via two logistics companies, A and B. $70\,\%$ of orders are sent with company A and the rest with company B. It has been observed that $5\,\%$ of orders from company A arrive late, whereas for company B, delays affect $10\,\%$ of packages.

1. a) Calculate the probability that an order arrives on time.

1. b) If a customer receives their order late, what is the probability that it was transported by company A?

In a digital urn in a game of chance there are 10 spheres: 6 blue and 4 white. The experiment consists of drawing two spheres successively and without replacement.

1. a) Draw the tree diagram detailing the probabilities on each branch.

1. b) Calculate the probability that both spheres are the same colour.

1. c) Calculate the probability that the first sphere is blue given that the second was white.

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

A Year 13 student has prepared 18 of the 25 topics for a History exam. The exam consists of choosing one topic out of two drawn at random by the teacher.

1. a) What is the probability that the student knows both topics drawn?

1. b) What is the probability that the student passes the exam (that is, knows at least one of the two topics)?

1. c) If the student has passed the exam, what is the probability that they knew only one of the two topics drawn?

In a secondary school, $40\,\%$ of the students are Science students and $60\,\%$ are Social Sciences students. $75\,\%$ of the Science students use school transport, whereas only $30\,\%$ of the Social Sciences students do so.

1. a) If we choose a student at random, what is the probability that they do not use school transport?

1. b) If we select a student who uses school transport, what is the probability that they are a Social Sciences student?

A wireless headphone factory produces two models: "Basic" and "Pro". $65\,\%$ of production corresponds to the Basic model. Quality control detects that $2\,\%$ of Basic headphones are defective, whereas for the Pro model the percentage of defective units is $1\,\%$.

1. a) Calculate the probability that a pair of headphones chosen at random is defective.

1. b) If a pair of headphones is defective, which model is it more likely to be? Justify your answer by calculating the posterior probabilities.

In the sorting center of an e-commerce company in Madrid, $90\,\%$ of packages are packed using a high-speed automated line and the remaining $10\,\%$ via a manual line for irregularly shaped items. The automated line has a sealing defect in $2\,\%$ of shipments, while the manual line has a sealing defect rate of $18\,\%$.

Before going out for delivery, a sensor detects a package with a sealing defect.

1. a) Estimate and reason before calculating: At first glance, do you think it is more likely that this defective package comes from the automated line or from the manual line? Justify your intuition by weighing two opposing factors: the large difference in the volume of packages processed by each line versus the difference in their error rates.
2. b) Draw the complete tree diagram modeling the situation, clearly indicating the first-level partition and the second-level conditional probabilities.
3. c) Calculate the total probability that a package chosen at random at the center has a defective seal.
4. d) Formally verify your estimate: calculate the probability that the defective package comes from the automated line. Compare the exact numerical result with your answer in part a) and explain which bias usually leads to mistakes in this type of situation.

According to data from the Ministry of Education and Vocational Training on university entrance exams (EvAU) in an autonomous community, $60\,\%$ of enrolled students come from the Science and Technology track, and $40\,\%$ from the Humanities and Social Sciences track. In the ordinary session, $95\,\%$ of Science students pass the compulsory general phase, while in Humanities and Social Sciences $90\,\%$ pass it.

Among those who pass the general phase, $35\,\%$ of Science students and $20\,\%$ of Humanities and Social Sciences students achieve a final grade above $10$ points (out of $14$) thanks to the weighted subjects of the voluntary phase.

1. a) Model the academic path of a randomly chosen student using a three-level tree diagram: Bachillerato track, passing the general phase, and obtaining a final grade above $10$ points. Write the corresponding probability in decimal notation on each branch.
2. b) Calculate the probability that a student chosen at random from that community passes the general phase and also obtains a grade above $10$ points.
3. c) Identify the path in the tree that describes a Humanities and Social Sciences student who passes the general phase but does not exceed $10$ points. Calculate its probability and interpret what percentage of the total Bachillerato student body this specific profile represents.

On an e-sports platform, players are classified according to their rank into: Bronze (50%), Silver (30%) and Gold (20%). It has been observed that the probability of a player cheating (using banned software) depends on their rank: in Bronze it is 2%, in Silver 5% and in Gold 10%. The FairPlay security system correctly detects a cheater 98% of the time, but gives a "false positive" (accuses a clean player) in 1% of cases.

1. a) Draw the tree diagram representing the situation of a player chosen at random undergoing the security check.
2. b) Calculate the probability that the FairPlay system issues a "cheating player" alert.
3. c) If the system has issued an alert, what is the probability that the player is actually innocent and of Gold rank?

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

A courier company uses three routes to deliver packages in a difficult city. Route A (used 40% of the time) has a 15% probability of delay. Route B (used 35% of the time) has a 10% probability of delay. Route C is used the rest of the time and has a 20% probability of delay. If a package arrives late, an audit protocol is applied to it. This protocol fails and does not detect the real reason for the delay in 5% of cases for route A, in 8% for route B and in 12% for route C.

1. a) Represent using a tree diagram the probability that a package chosen at random experiences a delay and its audit fails.
2. b) Calculate the probability that a package selected at random has a failed audit due to a delay.
3. c) Given that the audit has failed, what is the probability that the package did not travel via route A?

Three urns are available with the following composition of coloured balls:

• Urn $U_1$: 5 white, 3 black and 2 red.

• Urn $U_2$: 4 white and 6 black.

• Urn $U_3$: 2 white, 5 black and 3 red.

The experiment consists of rolling a fair die. If a prime number is rolled, a ball is drawn from $U_1$. If a composite number is rolled, a ball is drawn from $U_2$. In any other case, it is drawn from $U_3$. After the draw, the ball is placed into the urn from which it was not drawn that has the most white balls at that moment (in the event of a tie, $U_3$ is chosen). Finally, a second ball is drawn from that last urn.

1. a) Construct the detailed tree diagram of the two draws.
2. b) What is the probability that the second ball drawn is red?
3. c) If the second ball was red, what is the probability that the first ball was white?

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

A study on sleep habits among teenagers reveals that 25% sleep less than 6 hours, 60% sleep between 6 and 8 hours, and the rest sleep more than 8 hours. The probability of lacking concentration in the first class of the morning is 0.8 for the first group, 0.3 for the second and 0.1 for the third. Furthermore, if a student lacks concentration, the probability that they forget to hand in their homework is 0.7, whereas if they are focused, that probability drops to 0.05.

1. a) Calculate the probability that a student chosen at random forgets to hand in their homework.
2. b) If a student has handed in their homework correctly, what is the probability that they belong to the group that sleeps more than 8 hours?

In a wind turbine factory, 1% of blades have structural microcracks. An ultrasound scanning system is used that detects a crack with a probability of 0.99. However, the scanner has a 0.03 probability of indicating a crack when the blade is undamaged. To ensure quality control, if the scanner indicates a "crack", a second independent manual test is performed that detects an actual crack with a probability of 0.95 and has a 1% margin of error on undamaged blades.

1. a) Calculate the probability that a blade undergoes both tests and is declared "defective".
2. b) If a blade has been declared "defective" after both tests, what is the probability that it actually has no microcracks?

Three friends, Hugo, Silvia and Clara, play a turn-based strategy game. The probability of Hugo winning is $P(H) = 0,2$, Silvia's is $P(S) = 0,5$ and Clara's is $P(C) = 0,3$. After winning a game, the probability of the same player winning the next one increases by 20% relative to their initial probability (subtracting that difference proportionally from the other two players).

1. a) If Silvia wins the first game, construct the probability tree for the second game.
2. b) Calculate the probability that Silvia wins the first two games.
3. c) Given that Silvia won the second game, what is the probability that Clara won the first game?

There is a box containing 10 smartphones, 3 of which have a defective screen. Two phones are drawn at random, one after another and without replacement.

• If the first phone drawn is defective, a biased coin is tossed where the probability of heads is 0.7. If it lands on heads, a third phone is drawn from the box.

• If the first phone drawn is not defective, a fair die is rolled. If a number greater than 4 is rolled, a third phone is drawn.

1. a) Calculate the probability that the total number of defective phones drawn is exactly 2.
2. b) Given that at the end of the experiment exactly 2 defective phones have been drawn, what is the probability that the first phone drawn was non-defective?

A technology company produces processors in three plants: P1 (produces 50%), P2 (30%) and P3 (20%). The percentage of processors exceeding a speed of 4 GHz is 80% in P1, 70% in P2 and 60% in P3. Processors that do not exceed this speed undergo an optimisation process that results in 40% of them ultimately exceeding it.

1. a) Using a tree diagram, calculate the probability that a processor chosen at random from total production ultimately exceeds 4 GHz.
2. b) If a processor does not exceed 4 GHz after the whole process, what is the probability that it comes from plant P1 or P2?

In an electronic waste processing and recycling plant, integrated circuits are automatically diverted into two sorting lines: line $A$ and line $B$. The proportion of components diverted to line $A$ is an unknown value $p$ (with $0 < p < 1$), with the remainder ($1 - p$) diverted to line $B$.

Technical records show that a component processed on line $A$ is directly reusable with a probability of $0,75$. In contrast, on line $B$, the probability of direct reuse is only $0,35$.

All components that are not directly reusable move to a second stage of chemical refining to extract precious metals: on line $A$, $20\,\%$ of non-reusable components yield precious metals, whereas on line $B$, precious metals are recovered from $40\,\%$ of non-reusable components.

1. a) Draw the complete tree diagram representing the stages of the process, leaving the probabilities of the first branches in terms of the parameter $p$.
2. b) Obtain the simplified algebraic expression, in terms of $p$, representing the probability that a randomly chosen integrated circuit ends up yielding precious metals after failing direct reuse.
3. c) Knowing that at the end of the day exactly $13\,\%$ of all components received at the plant end up providing precious metals through chemical refining, set up an equation and deduce the original value of the proportion $p$.
4. d) Justify analytically whether it would be possible, solely by modifying the initial input distribution $p$, to achieve an overall percentage of components yielding precious metals equal to $30\,\%$.

A pharmaceutical quality control laboratory must validate a small batch of $15$ units of a biological reagent. It is known with certainty that in the batch there are exactly $3$ defective units and $12$ non-defective ones. Due to the cost of the procedure, a destructive sequential inspection is applied (the tested units become unusable and, therefore, are sampled without replacement), governed by the following protocol:

• A first unit is drawn at random: if it turns out to be defective, the batch is rejected and the process is stopped immediately.

• If the first unit is non-defective, a second unit is drawn. If this second unit is defective, a third unit must be drawn as a tie-breaker; on the other hand, if the second unit is non-defective, the batch is declared accepted and the process ends immediately.

• If the third draw has to be reached, the batch is declared accepted if that third unit is non-defective, and permanently rejected if it is defective.

1. a) Model the entire protocol using a tree diagram. Keep in mind that the branches have different lengths (a truncated tree depending on the outcome) and that the batch composition changes after each draw. Express the probabilities of each branch as irreducible fractions.
2. b) Reasonably justify why, in this experiment, it would be a serious error to calculate the branch probabilities assuming independence between successive stages.
3. c) Calculate the total probability that the batch ends up being declared accepted.
4. d) Imagine that the test were non-destructive and the draws were carried out with replacement (returning each unit after testing it). Reason, without needing to calculate all the decimals, whether the probability of declaring the batch accepted in exactly two draws would be greater than, less than, or equal to the case without replacement.