Tree diagrams and contingency tables

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?

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.

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.

In a study on the use of social media at a secondary school in Madrid, it has been observed that, in a group of $100$ Bachillerato 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 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?

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.