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?

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 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.

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 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.

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 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 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 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 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 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?

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 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?

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 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.