Exercises: Tree diagrams

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

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

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

A TikTok content creator observes that $70 \%$ of their videos are comedy and $30 \%$ are challenges. The probability that a comedy video goes viral is $0,1$, while for challenge videos this 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 high school senior has prepared $60 \%$ of the topics for a Geography exam. The exam consists of drawing a topic at random from a bag. If the drawn topic is one they have studied, the probability of passing is $0,9$. If the topic is one they have not studied, the probability of passing drops to $0,2$.

1. a) Draw a tree diagram with the data from the problem.

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 $60 \%$ a "Free" account. It is known that $80 \%$ of Premium users listen to music daily, while only $30 \%$ of users with a free account do so. Choosing a user at random:

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

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

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

1. a) Construct the tree diagram.

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

In an urn we have $4$ red balls and $6$ blue balls. We draw two balls successively without replacement (without putting the first ball back into the urn).

1. a) Represent the process using a tree diagram indicating the probabilities on each branch. Bear in mind that the composition of the urn changes after the first draw.

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

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

1. a) Represent the situation with a probability tree.

1. b) Calculate the probability that a hoodie chosen at random has no defects.

A popular video game streamer broadcasts content on three different platforms: Twitch, YouTube and TikTok Live. According to their statistics, $45\,\%$ of their followers watch their 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 application uses a smart sensor to detect whether a user is doing intense aerobic exercise. The sensor is able to correctly detect the activity $98\,\%$ of the time. However, when the user is resting or walking slowly, the sensor gives a "false positive" (erroneously indicates that they are doing intense exercise) in $3\,\%$ of cases. According to data from the user community, it is estimated that they spend $15\,\%$ of their time doing intense activity.

1. a) What is the probability that, at a randomly chosen moment, the application indicates that the user is doing intense exercise?

1. b) If the application indicates that the user is doing intense exercise, calculate the probability that they are actually resting 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 they choose the Warrior, they have a $0,7$ probability of winning the match. If they choose the Mage, their probability of victory increases to $0,85$.

1. a) Construct the tree diagram associated with the experiment.

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 they chose the Warrior?

An online clothing store sends its orders via two logistics companies, A and B. $70\,\%$ of the orders are sent with company A and the rest with company B. It has been observed that $5\,\%$ of company A's orders arrive late, whereas in company B the delay affects $10\,\%$ of the parcels.

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 of 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 the two spheres are the same color.

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

A high school senior has prepared 18 of the 25 topics for a History exam. The exam consists of choosing one topic from two drawn at random by the teacher.

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

1. b) What is the probability that the student passes the exam (that is, they know 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 drawn topics?

In a high school, $40\,\%$ of the students are in Science and $60\,\%$ in Social Sciences. $75\,\%$ of the Science students use school transport, while only $30\,\%$ of the Social Science 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 in Social Sciences?

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

1. a) Calculate the probability that a headphone chosen at random is defective.

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

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 prohibited software) depends on their rank: in Bronze it is 2%, in Silver it is 5% and in Gold it is 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 randomly chosen player going through 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?

A courier company uses three routes to deliver parcels in a difficult city. Route A (used 40% of the time) has a 15% probability of delay. Route B (used 35%) has a 10% delay. Route C is used the rest of the time and has a 20% delay. If a parcel 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, 8% for route B and 12% for route C.

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

There are three urns with the following composition of colored 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 for 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 was white?

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 morning class 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 of forgetting to hand in their homework is 0.7, whereas if they are concentrated this probability drops to 0.05.

1. a) Calculate the probability that a randomly chosen student 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 the blades have structural micro-cracks. An ultrasound scanning system is used which detects the crack with a probability of 0.99. However, the scanner has a probability of 0.03 of indicating that there is a crack when the blade is sound. To ensure control, if the scanner indicates "crack", a second independent manual test is carried out which detects the real crack with a probability of 0.95 and has a 1% margin of error on sound blades.

1. a) Calculate the probability that a blade goes through both tests and is declared "defective".
2. b) If a blade has been declared "defective" after the two tests, what is the probability that it actually has no micro-cracks?

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

1. a) If Chloe wins the first game, construct the probability tree for the second game.
2. b) Calculate the probability that Chloe wins the first two games.
3. c) If we know that Chloe has won the second game, what is the probability that Sophia won the first?

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

• If the first one 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 one 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 fine?

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

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