Tree Diagrams

A box contains $4$ red balls and $6$ blue balls. A ball is drawn at random, its colour is noted, it is returned to the box, and a second ball is drawn. Draw the corresponding tree diagram and calculate the probability of obtaining two blue balls.

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

Complete the following tree diagram representing the tossing of a biased coin twice consecutively, given that the probability of getting heads is $\frac{1}{3}$. From the tree diagram, calculate the probability of getting heads first and then tails.

A basketball player has a probability of $0.8$ of scoring a free throw. If they take two free throws independently, draw the probability tree diagram and determine the probability that they miss both shots.

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

A café offers two types of set menu: $60\,\%$ of the customers choose the set menu of the day and $40\,\%$ choose the vegetarian menu. Of those who choose the menu of the day, $70\,\%$ order coffee for dessert. Of those who choose the vegetarian menu, $80\,\%$ order tea. Calculate the probability that a randomly chosen customer ordered the menu of the day and also ordered coffee.

In a wardrobe there are $5$ white T-shirts and $3$ black T-shirts. Two T-shirts are chosen at random, one after the other and without replacing the first back into the wardrobe. Calculate the probability that both T-shirts are white.

A Spanish pack of $40$ cards is available. Three cards are drawn consecutively without replacement. Calculate the probability that all three cards are of the same suit (coins, cups, swords or clubs).

In an electronic components factory, three machines (A, B, and C) produce $50\,\%$, $30\,\%$, and $20\,\%$ of the parts, respectively. The percentages of defective parts produced by each machine are $2\,\%$, $4\,\%$, and $5\,\%$. Use the tree diagram to calculate the probability that a randomly chosen part is defective.

We have two urns. Urn A contains $3$ red balls and $2$ black balls. Urn B contains $4$ red balls and $5$ black balls. We toss a coin: if it comes up heads, we draw a ball from urn A; if it comes up tails, we draw a ball from urn B. What is the probability that the drawn ball is red?

In a diagnostic test for an allergy, $5\,\%$ of the population actually has the allergy. If a person has the allergy, the test is positive $90\,\%$ of the time. If they do not have it, the test is negative $95\,\%$ of the time. Draw a tree diagram to illustrate the situation and calculate the probability that a randomly chosen person gets a positive result on the test.

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

A vault alarm system has 3 motion sensors. The probability that sensor A fails during an intrusion is $0.1$; that sensor B fails is $0.05$; and that sensor C fails is $0.2$. The sensors act completely independently. Draw the corresponding tree diagram and calculate the probability that, during an actual intrusion, at least one sensor activates and sounds the alarm.

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

An urn contains $5$ red balls and $3$ white balls. Three balls are drawn successively at random and without replacement. The goal is to calculate the probability of obtaining at least one white ball.\nThree students propose different strategies using trees:\n- Seán proposes drawing the complete tree of the three draws ($2^3 = 8$ final branches), identifying all paths containing one, two, or three white balls, and adding their probabilities.\n- Aoife proposes calculating the complementary event («no white balls», that is, three consecutive red balls) using a single branch of the tree and subtracting its probability from $1$.\n- Cian proposes a «pruned tree»: if a white ball is drawn on the first or second draw, he stops that branch (it is already a success), and only branches out when a red ball is drawn.\na) Develop and calculate the requested probability using both Aoife's and Cian's methods, verifying that both lead to the exact same irreducible fraction.\nb) Rigorously compare the three methods: indicate how many paths and multiplications each one requires you to calculate, and justify why Seán's method is the least efficient and the most prone to algebraic errors.\nc) If the problem asked to «obtain exactly two white balls», would Aoife's complementary event method still be advantageous? Justify which method would be optimal in that case.

In a city, $20\,\%$ of drivers are young (under 25 years old) and $80\,\%$ are adults. The probability that a young driver has an accident in a year is $0.15$, whereas for an adult it is $0.05$. A driver is selected at random and turns out to have had an accident this year. Using the branches of a tree diagram, calculate the probability that this driver is young. (Express the result as an irreducible fraction).