Exercises: Trees

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.

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.

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.

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.

We roll a six-sided die and then draw a card from a Spanish pack of $40$ cards. If an even number comes up on the die, we win if we draw a King. If an odd number comes up on the die, we win if we draw an Ace. Calculate the probability of rolling an even number on the die and also winning.

To go to work, Yasmin can go by bus (with a probability of $0,7$) or by bicycle (with a probability of $0,3$). If she goes by bus, the probability of arriving late is $0,2$. If she goes by bicycle, the probability of arriving late is $0,05$. What is the probability that Yasmin goes by bicycle and arrives on time?

An urn contains $2$ green balls and $3$ yellow balls. We draw a ball, look at it, and leave it out. Then we draw another ball. What is the probability that the first ball is green and the second is yellow?

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.

In a bag there are $3$ black counters and $2$ white ones. Two counters are drawn successively and without replacement. To calculate the probability that the second counter is white, Lucas draws a tree diagram. In the second stage, he observes two branches ending in "white": one with probability $\frac{1}{4}$ (if the first was white) and another with probability $\frac{2}{4}$ (if the first was black). Lucas concludes: "Since both branches end in white, the total probability that the second counter is white is $\frac{1}{4} + \frac{2}{4} = \frac{3}{4} = 0,75$".\na) Explain in detail the conceptual error in reasoning that Lucas makes when directly adding those two branches of the second level.\nb) Correct his calculation by properly using the probabilities of the complete paths of the tree and compare the result with the probability of having drawn white on the first draw. What conclusion do you reach?

In the design of an adventure video game, a chest can be trapped (with probability $0,3$) or contain a reward (with probability $0,7$). If it is trapped, the probability of the hero successfully disarming it is $0,2$. If it contains a reward, it may be intact or deteriorated over time.\nTo balance the difficulty, the developer needs the total probability of the player «obtaining an intact reward or successfully disarming the trap» to be exactly $0,62$.\na) Draw the tree diagram of the situation, identifying the stages and branches, and denoting by $p$ the conditional probability that «the reward is intact given that there was no trap».\nb) Set up and solve an equation using the branches of the tree to find the exact value that $p$ must have.

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.

In a tennis match, if a player's first serve is in (probability $0,6$), they win the point with probability $0,7$. If the first serve misses, they take a second serve which is in with probability $0,8$. If the second serve is in, they win the point with probability $0,5$. If the second serve also misses, they commit a double fault and automatically lose the point. Calculate the total probability that the player wins the point on that service.

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?

A student has to take two exams on the same day. The probability of passing the first is $0,6$. If they pass the first, they get motivated and the probability of passing the second rises to $0,8$. However, if they fail the first, they get discouraged and the probability of passing the second drops to $0,3$. Construct the tree diagram and calculate the probability that they pass at least one of the two exams.

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

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 bag contains $5$ white balls and $4$ black balls. Balls are drawn one by one without replacement until a black ball is drawn. What is the probability of having to carry out exactly three extractions for the game to end?

The weather in a coastal city follows a curious pattern: if a day is sunny, the probability that it rains the next day is $0,2$. If a day is rainy, the probability that it keeps raining the next day is $0,6$. Given that today is Monday and it is sunny, what is the probability that it rains on Wednesday? Draw the corresponding tree diagram for Tuesday and Wednesday.

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

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).

According to AEMET records in a northern town, it rains on $10\,\%$ of days in November. Data from the municipal transport company indicate that, if it rains, the probability of a bus line experiencing a significant delay is $0,40$; however, if it does not rain, that probability decreases to $0,05$.\na) Before calculating anything, estimate intuitively: if on any given day in November that line experiences a significant delay, do you think it is more likely that it rained or that it did not rain? Briefly justify your intuition considering the weight of each type of day.\nb) Model the situation with a tree diagram and calculate the total probability that the bus experiences a significant delay on a day in November.\nc) Calculate using Bayes' theorem the probability that it rained given that the bus arrived late. Check whether your initial estimate in part a) was correct or if you succumbed to the «base rate fallacy».

According to DGT data on practical driving tests, in a certain province $45\,\%$ of candidates pass on their first attempt. Those who fail take a second test after completing refresher driving lessons, with $60\,\%$ of them managing to pass on this second attempt. Those who fail the second test do not continue that year.\na) Model the entire process using a tree diagram, indicating at each node and branch the events («passes on 1st», «fails on 1st», «passes on 2nd», «fails on 2nd») and their associated probabilities.\nb) What total percentage of candidates obtain their driving licence in a maximum of two attempts?\nc) If you randomly select a person who has obtained their licence through this process, what is the probability that they achieved it on their first attempt? Interpret what this value means for managing waiting lists at the driving school.

There are two boxes. Box 1 contains $4$ blue counters and $2$ red counters. Box 2 contains $3$ blue counters and $5$ red counters. A counter is drawn at random from Box 1 and placed into Box 2. Next, a counter is drawn from Box 2. Calculate the probability that the counter drawn from Box 2 is blue.

An urn contains $x$ white balls and $4$ black balls. Two balls are drawn without replacement. Knowing that the probability that both balls are white is $\frac{1}{3}$, set up the corresponding equation with the help of a tree diagram and determine the initial number of white balls $x$.

Two e-sports teams, Alpha and Beta, play the final of a best-of-three tournament (the first to win two matches is the champion). Alpha is the better team, so their probability of winning any match is $0,6$. Matches cannot end in a draw. Draw the tree diagram of all possible outcomes of the final and calculate the probability that Team Beta wins the tournament.

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.

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).

In a board game, a player must cross a pit by jumping on three consecutive platforms. The probability of slipping on the first platform is $\frac{1}{10}$. If they pass the first, the probability of slipping on the second is $\frac{1}{5}$. If they pass the second, the probability of slipping on the third is $\frac{1}{4}$. If they slip on any platform, they fall into the pit and lose. What is the probability that the player crosses the pit successfully? Given that the player fell into the pit, what is the probability that they fell on exactly the second platform?

An urn contains $5$ balls numbered from $1$ to $5$. Balls are drawn one by one, without replacement, until the sum of the numbers drawn is greater than or equal to $6$. Draw the tree diagram with all possible branches, given that the ball with number $2$ was drawn on the first extraction. From your tree diagram, calculate the probability that the game ends on exactly the third extraction.

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

In a competition, an archer has a maximum of three arrows to hit the target. The probability of hitting the target with the first arrow is $0,4$. If they miss, they adjust their sight and the probability of hitting with the second is $0,6$. If they miss again, they get nervous and the probability of hitting with the third is $0,3$. The archer stops shooting as soon as they hit the target. Calculate the probability that the archer needed to take more than one shot given that they ultimately managed to hit the target.

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- Álvaro 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- Beatriz 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- Carlos 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 Beatriz's and Carlos'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 Álvaro'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 Beatriz's complementary event method still be advantageous? Justify which method would be optimal in that case.

A security filter analyses incoming emails for a banking institution. It is known that $10\,\%$ of received emails are fraudulent (phishing) and $90\,\%$ are legitimate. The detection system correctly classifies $95\,\%$ of emails that are phishing as a threat. However, it is not infallible with legitimate emails: there is an unknown probability $p$ of a «false positive» (a legitimate email that the filter mistakenly flags as a threat).\nTo avoid overwhelming the cybersecurity team with false alerts, management requires that, when the filter triggers an alert, the probability of the email actually being phishing is at least $90\,\%$.\na) Model the problem using a tree diagram, leaving the probability $p$ indicated on the branch corresponding to false positives.\nb) Find the algebraic expression, in terms of $p$, for the probability that an email is phishing given that the filter has triggered an alert.\nc) Set up and solve an inequality to determine the maximum allowable value of $p$. Explain why, even though the filter detects $95\,\%$ of phishing, the false positive rate $p$ must remain at an extraordinarily low value.

Topic Challenge. Does yesterday's weather predict today's? You are going to check with real data whether rainfall in your town exhibits dependent or independent events.

1. Data collection: Go to the AEMET website and record in a spreadsheet the daily precipitation in your municipality over the last $30$ days. Define the simple event $L$: "rainy day" ($>0,1$ mm).

2. Analysis: Calculate the probability of $L$ and of its complement. Count pairs of consecutive days to calculate the conditional probabilities $P(L_2|L_1)$ and $P(L_2|\overline{L}_1)$. Compare the values with $P(L)$ using the independence condition $P(L_2|L_1)=P(L)$ and reason whether your sample suggests dependent or independent events (this is a provisional estimate: there are only $30$ days; if there is no rainy day or no dry day, extend the period).

3. Modeling: Build a tree diagram with these probabilities and calculate the probability of the compound event "raining two days in a row this weekend".

4. Product: Prepare a short one-page report with the linked spreadsheet, the solved tree, and a reasoned conclusion about the reliability of predicting weather using this model.

Assessment criteria:

• Correct calculation of simple and conditional probabilities from real AEMET data.

• Reasoned mathematical justification of the dependence or independence observed in the sample, acknowledging its provisional nature.

• Proper construction of the tree diagram and resolution of the compound event.

• Clarity of presentation and correct use of the spreadsheet and probability notation.