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.

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.

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?

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?

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 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?

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?

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?

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?

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?