Exercises: Dependent events

In a box there are $8$ LED light bulbs, $3$ of which are blown. If two bulbs are drawn at random, one after the other without replacement, calculate the probability that both work correctly.

From a Spanish pack of $40$ cards, two cards are drawn consecutively without replacement. Determine the probability that the first card is a king and the second card is an ace.

In a class of $30$ students, $18$ are girls. It is known that $12$ girls and $5$ boys study French. If a student is selected at random from this class and turns out to be a girl, calculate the probability that she studies French.

Let $A$ and $B$ be two events of a random experiment such that $P(A) = 0,4$, $P(B) = 0,5$ and $P(A \cap B) = 0,25$. Check mathematically whether events $A$ and $B$ are dependent or independent.

In a drawer there are $5$ black socks and $3$ blue socks, all mixed together. If you take out two socks in the dark, one after another, calculate the probability that both are blue.

The following tree diagram shows the probabilities of a basketball player scoring ($E$) or missing ($F$) their first and second free throws. Observe the conditional probabilities and calculate the probability that the player misses the first throw and scores the second.

In an animal shelter there are dogs and cats. The probability of choosing a dog at random is $0,6$. If we know that the chosen animal is a dog, the probability that it has a collar is $0,8$. What is the probability of choosing an animal at random that is a dog and also has a collar?

From an urn containing $4$ red balls and $6$ green balls, three balls are drawn consecutively without replacement. Calculate the probability that all three balls drawn are red.

Lucía is solving a classroom problem: "In a Spanish deck of $40$ cards, two consecutive cards are drawn without replacement. What is the probability that both are cups?". Lucía writes in her notebook:

"There are $10$ cups in the deck, so the probability of drawing a cup the first time is $\frac{10}{40}$. Since the event does not change type on the second draw, the probability of drawing a cup again is once more $\frac{10}{40}$. Therefore, the probability of drawing two cups is $\frac{10}{40} \cdot \frac{10}{40} = \frac{100}{1600} = \frac{1}{16} = 0,0625$".

1. a) Point out precisely what conceptual error Lucía made by considering that the two events are independent.
2. b) Explain how the fact of not replacing the first card affects both the number of favorable outcomes and the number of possible outcomes on the second draw.
3. c) Calculate the correct probability as an irreducible fraction and as a decimal number.

In a transport inspection carried out by the Dirección General de Tráfico (DGT) in a region of Castilla y León, officers stop a group of $15$ delivery vans. After reviewing the documentation in the central database, it is found that exactly $3$ of them are driving with an expired Inspección Técnica de Vehículos (ITV). If the officers randomly select $2$ vans from that group, one after the other and without replacement:

1. a) Model the situation by identifying the events of each stage and state why the probability for the second van depends on what happened with the first one.
2. b) Calculate the probability that both selected vans have their ITV fully up to date. Express the result as an irreducible fraction and as a percentage rounded to two decimal places.

A batch of $50$ mobile phones contains $4$ units with manufacturing defects. A quality inspector selects $3$ phones at random from the batch, one by one and without replacement. Calculate the probability that none of the selected phones is defective.

In a survey of $200$ people about their preference between tea and coffee, the following data were obtained: $120$ prefer coffee, of whom $70$ are women. If there are $110$ women in total in the survey, calculate the probability that a person chosen at random prefers tea, given that they are a man.

We have two urns and a fair six-sided die. We roll the die: if a number less than $3$ comes up, we draw a ball from Urn $A$; otherwise, we draw a ball from Urn $B$. Observe the contents of the urns in the image and calculate the probability that the ball drawn is red.

A student has a keyring with $5$ very similar keys, only one of which opens his front door. If he arrives at night and tries the keys at random, discarding those that do not work, what is the probability that he manages to open the door on exactly the third attempt?

The probability of rain in a city on a given day depends on the weather the previous day. If it rains today, the probability that it rains tomorrow is $0,7$. If it does not rain today, the probability that it rains tomorrow is $0,2$. Given that today is Monday and it is raining, calculate the probability that it will not rain on Wednesday.

In a board game, a chest contains $6$ gold coins and $4$ silver coins. A player draws two coins without looking, one after another without replacement. Based on the tree diagram outline, calculate the probability that the two coins drawn are of different materials.

In a hospital, $40\%$ of the patients are admitted for respiratory diseases. Of these, $15\%$ are smokers. Among patients admitted for other reasons, $20\%$ are smokers. If a patient is chosen at random and turns out to be a smoker, what is the probability that they are admitted for a respiratory disease?

From a Spanish pack of $40$ cards, we draw two cards at random without replacement. Calculate the probability that at least one of the cards drawn is a knight.

According to historical climate data from the Agencia Estatal de Meteorología (AEMET) for the month of November in a city on the Cantabrian coast, the weather shows strong persistence: if it rains on a given day, the probability that it rains the following day is $0,70$; whereas if it does not rain on a given day, the probability that it rains the following day drops to $0,30$. We know for certain that it rained heavily on Tuesday.

1. a) Before doing any numerical calculations, give a reasoned estimate of whether the probability of it raining on Thursday (two days after Tuesday) will be higher or lower than $50\,\%$ ($0,50$).
2. b) Draw a tree diagram showing the possible paths for Wednesday and Thursday, calculate the exact probability that it rains on Thursday, and check whether your initial estimate was correct, explaining the reason for the result obtained.

At a health centre in a provincial capital, the records of $1000$ patients who attended with a fever during a winter campaign are analysed. Out of the total number of patients, $200$ actually had the influenza virus and $800$ suffered from other common viral infections. The rapid test used in triage correctly detects the presence of the virus in $90\,\%$ of people who have the flu, but produces a false positive in $10\,\%$ of people who do not have it. We want to find the probability that a randomly chosen patient tests positive.

To solve this, two students suggest different methods:

• Method 1 (Tree diagram and total probability): Assign probabilities to each branch ($P(\text{Flu})$, $P(\text{Positive}\mid\text{Flu})$, etc.) and add up the paths leading to a positive test.

• Method 2 (Contingency table with absolute frequencies): Fill in a $2\times 2$ table with the exact number of people in each category and divide the total number of positive tests by $1000$.

1. a) Apply both methods to calculate this probability and verify that they yield exactly the same value.
2. b) Compare both approaches: which one do you consider clearer and less prone to errors when explaining it to non-specialists in mathematics? Justify your answer by weighing the pros and cons of each.

An urn contains $x$ white balls and $5$ black balls. If two balls are drawn at random without replacement, the probability that both are white is $\frac{1}{6}$. Set up the corresponding equation and determine the initial number of white balls in the urn.

A screening test for a disease has a reliability of $95\%$ for detecting sick people and $90\%$ for testing negative in healthy people. It is known that $2\%$ of the population suffers from this disease. If a person takes the test and the result is positive, calculate the probability that they are actually sick.

Two urns are available with the contents shown in the figure. A ball is drawn at random from Urn $I$ and, without looking at its colour, is placed into Urn $II$. Then, a ball is drawn from Urn $II$. Calculate the probability that the ball drawn from Urn $II$ is white.

In a chess tournament, a player faces the same opponent in a best-of-three series (the first to reach two wins wins). The probability of winning the first game is $0,5$. If he wins a game, his confidence increases and the probability of winning the next is $0,6$. If he loses, his probability of winning the next drops to $0,4$. Assuming there are no draws, calculate the probability that this player wins the tournament.

There are three boxes that are identical on the outside. Box $A$ contains $2$ gold coins, Box $B$ contains $2$ silver coins, and Box $C$ contains $1$ gold coin and $1$ silver coin. A box is chosen at random and a coin is drawn at random from it, which turns out to be gold. What is the probability that the other coin remaining in that same box is also gold?

An alarm system consists of two sensors, $A$ and $B$. Under normal conditions, sensor $A$ fails $5\%$ of the time and sensor $B$ fails $8\%$ of the time. However, due to joint electrical interference, the probability that sensor $B$ fails given that sensor $A$ has failed increases to $30\%$. Calculate the probability that, during an intrusion, the entire system fails (that is, both sensors fail).

From a Spanish pack of $40$ cards, three cards are drawn consecutively without replacement. Calculate the probability of drawing exactly two kings and one knight, taking into account that they can appear in any order.

In a technology company, $60\%$ of the employees are programmers, $30\%$ are designers and $10\%$ are managers. $80\%$ of the programmers, $40\%$ of the designers and $90\%$ of the managers speak fluent English. An employee is selected at random and found not to speak fluent English. Calculate the probability that this employee is a designer.

In a canning plant in the Rías Baixas (Galicia), mussel packaging is divided between two automated lines, $L_1$ and $L_2$. Line $L_1$ handles $60\,\%$ of daily production and line $L_2$ handles the remaining $40\,\%$. Quality audits indicate that the probability of a can from $L_1$ having a defective seal is $0,01$ ($1\,\%$). The technical management requires that the overall probability of finding a defective can in the entire day's production does not exceed $0,018$ ($1,8\,\%$).

1. a) Set up an inequality based on the law of total probability that allows you to determine the maximum rate of defective cans permitted on line $L_2$ to meet the factory standard.
2. b) Assuming that line $L_2$ operates right at that maximum admissible limit, if an inspector randomly picks a can from the final production and finds it to be defective, calculate the conditional probability that it comes from line $L_2$ and interpret which line contributes more defective cans to the final batch.

According to road safety estimates from the DGT for festive nights in urban areas, approximately $4\,\%$ of drivers on the road have consumed alcohol above the legal limit. Local police use a portable screening device at checkpoints: if a driver exceeds the limit, the device tests positive $95\,\%$ of the time; if the driver does not exceed it, the device produces a false positive in $5\,\%$ of the tests.

1. a) Model the situation using a tree diagram with all associated probabilities and calculate the probability that a driver chosen at random at that checkpoint tests positive on the portable test.
2. b) If a randomly selected driver tests positive on the screening test, calculate the actual probability that they truly exceed the legal alcohol limit.
3. c) In light of the percentage obtained in part b), explain mathematically why Spanish regulations prohibit issuing a penalty based solely on this test and make it mandatory to perform a second measurement with a certified evidential breathalyzer after a $10$-minute wait.

Topic challenge. Should we take the risk of organising an outdoor festival? You are going to evaluate the weather risk in your town by making decisions based on real data.

1. Data collection and events: Check the records of the last $30$ days for your municipality on the AEMET website. In a spreadsheet, record the days with rain ($A$) and the days with strong wind gusts ($B$). Define the sample space and find the probability of each event using Laplace's rule (for example, $P(A) = 6/30 = 0,20$).

2. Independence and dependence: Analyse whether it raining two days in a row are independent or dependent events. Calculate in your spreadsheet the empirical compound probability of it raining one day given that it rained the previous day, $P(L_2 | L_1)$, and compare it with $P(L_2)$ to determine whether prior weather has an influence.

3. Decision report: Prepare a one-page report with a comparative table calculating the probability of it raining throughout an entire weekend under the hypothesis of independent versus dependent events. Add a reasoned proposal on whether the organisation should rent a marquee or take out cancellation insurance.

Assessment criteria:

• Accurate identification of events and correct calculation of probabilities using Laplace's rule.

• Justified mathematical distinction between dependent and independent events based on the data.

• Organisation of data in a spreadsheet and clarity of argument in the final proposal.