Exercises: Dependent and independent events

Let $A$ and $B$ be two events associated with a random experiment. We know that the probability of event $A$ occurring is $P(A) = 0,6$ and the probability of event $B$ occurring is $P(B) = 0,3$. If the events are known to be independent, calculate the probability of both occurring at the same time, that is, $P(A \cap B)$.

In a gumball machine there are balls of three colours: 10 red, 6 blue and 4 green. A child draws a gumball, notes its colour, puts it back into the machine and draws a second gumball.

1. a) Justify whether the events "drawing a red ball on the first" and "drawing a blue ball on the second" are dependent or independent.
2. b) Calculate the probability that the first gumball is blue and the second is green.

We draw two cards from a Spanish 40-card deck, one after the other and without returning the first to the deck.

1. a) What is the probability that the first card is a King?
2. b) Given that the first card was a King, what is the probability that the second is also a King?
3. c) Are these events dependent or independent?

A 1st Baccalaureate student has two exams on the same day: Mathematics and English. The probability of passing Mathematics is $P(M) = 0,7$ and that of passing English is $P(I) = 0,8$. Assuming that passing one exam does not affect the result of the other:

1. a) Calculate the probability of passing both exams.
2. b) Calculate the probability of passing neither exam.

In a drawer in a changing room of a clothes shop, 4 black socks and 2 white socks of the same size and feel have been left behind. If a shop assistant reaches in and draws two socks at random, one after the other and without looking:

1. a) Complete the following tree diagram with the corresponding probabilities.
2. b) Calculate the probability that both drawn socks are black.

A survey was carried out in a secondary school on the use of two social networks: Instagram ($I$) and TikTok ($T$). The results show that $70\%$ of the students use Instagram, $50\%$ use TikTok and $35\%$ use both platforms. Determine whether the use of Instagram is independent of the use of TikTok in this school.

In a quality control process at a light bulb factory, the probability of a light bulb being defective is known to be $0,02$. If two light bulbs are chosen at random, calculate the probability that:

1. a) Neither is defective.
2. b) Both are defective.

(Note: The batch size is so large that selecting a light bulb can be assumed not to affect the probability of the next).

Look at the following contingency table showing the preferred type of sport for a group of 100 teenagers:

A teenager is chosen at random. Calculate:

1. a) $P(F)$, the probability that they like football.
2. b) $P(F|H)$, the probability that they like football given that they are a boy.
3. c) In light of the results, are the events "being a boy" and "liking football" independent?

In a Mathematics Applied to the Social Sciences exam, the following question is posed: «We roll a fair six-sided die. We consider the events $A$: "rolling an even number" and $B$: "rolling an odd number". Are $A$ and $B$ independent?».\n\nLucas writes in his exam:\n«Since it is impossible to roll an even and an odd number at the same time, the events do not influence each other at all; therefore, they are independent because one cannot alter the other».\n\na) Find the conceptual error in Lucas's reasoning and explain in detail why he confuses the concepts of «mutually exclusive events» and «independent events».\nb) Correct the exercise by carrying out the rigorous formal mathematical verification using the equality $P(A \cap B) = P(A) \cdot P(B)$ to classify with certainty whether the events are dependent or independent.

In a small service cooperative of $10$ partner members, $4$ have a degree in international trade and $6$ do not. To attend a business meeting, $2$ different members are selected at random and consecutively (sampling without replacement). Let the events be $A$: «the first person selected has the degree» and $B$: «the second person selected has the degree».\n\na) Before calculating anything, intuitively estimate and reason: do you think that the occurrence of $A$ increases, decreases, or leaves unchanged the probability that $B$ occurs? Consequently, do you expect the events to be independent or dependent?\nb) Now calculate the exact values of $P(A)$, the conditional probability $P(B|A)$, and the probability of the intersection $P(A \cap B)$.\nc) Calculate the marginal probability $P(B)$ and numerically check whether the independence condition $P(A \cap B) = P(A) \cdot P(B)$ holds. Evaluate whether the mathematical result obtained matches your initial estimate and explain what role the lack of replacement played.

On a music streaming platform, $60\%$ of users listen to the "Urban" genre ($U$) and $45\%$ listen to "Pop" ($P$). It is also known that $27\%$ of users listen to both music genres.

1. a) Determine whether the events "listening to Urban" and "listening to Pop" are independent. Justify your answer mathematically.

1. b) If we select a user who we know listens to "Pop" music, what is the probability that they also listen to "Urban"?

An urn contains 5 yellow balls and 3 purple balls. We draw two balls from the urn consecutively and without replacement.

1. a) Draw the tree diagram corresponding to this experiment, indicating the probability of each branch.

1. b) Calculate the probability that the two drawn balls are the same colour.

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

A video game programmer is testing a character's "critical hit" system. The character has a probability of $0,2$ of landing a critical hit ($C$). However, the system is designed so that if the last hit was critical, the probability of the next hit also being critical drops to $0,1$ due to ability cooldown. If the last hit was not critical, the probability of the next hit remains at $0,2$.

1. a) Are the events "first hit critical" and "second hit critical" independent? Reason your answer.

1. b) Calculate the probability that, in a sequence of two hits, the character does not land any critical hits.

At a secondary school in Madrid, a study was conducted on breakfast habits and the school year of the students, obtaining the following data:

A student is chosen at random. Check whether the event "eating breakfast at home" is independent of the event "being in 1st Baccalaureate".

Two electronic components of a drone, $A$ and $B$, function independently. The probability of component $A$ failing is $0,05$ and the probability of component $B$ failing is $0,08$.

1. a) Calculate the probability that the drone can keep flying if at least one of the two components must function correctly for this to happen.

1. b) Calculate the probability that exactly one of the two components fails.

We draw two cards from a Spanish 40-card deck, one after the other and without replacing the first.

1. a) What is the probability that the second card is a King, given that the first card was the Ace of Coins?

1. b) What is the probability of obtaining two face cards (Jack, Knight or King) of different suits?

A Baccalaureate student has two independent alarm clocks on their mobile phone. The first ($D_1$) works on $95\%$ of mornings and the second ($D_2$) works $90\%$ of the time.

1. a) Calculate the probability that the student oversleeps because neither alarm clock goes off.

1. b) If we know that at least one of the alarm clocks went off, what is the probability that alarm clock $D_1$ failed?

Let $A$ and $B$ be two events in a sample space such that $P(A) = 0,4$, $P(B) = 0,3$ and $P(A \cup B) = 0,58$.

1. a) Calculate $P(A \cap B)$.

1. b) Determine whether events $A$ and $B$ are independent.

1. c) Calculate the conditional probability $P(A | \bar{B})$, where $\bar{B}$ is the complementary event of $B$.

At a provincial Traffic headquarters, data from $200$ candidates taking the practical driving test are analysed. Of the total number of candidates, $120$ previously took an intensive traffic simulator course ($S$) and the rest did not. After evaluating the records, it is found that $90$ of those who took the intensive course passed on the first attempt ($A$), whereas of those who did not take the intensive course, only $40$ passed.\n\na) Model the given information by constructing a complete contingency table of absolute frequencies and calculate the probabilities $P(S)$, $P(A)$, and $P(S \cap A)$.\nb) Formally determine whether the event «passing on the first attempt» and the event «having taken the simulator course» are mathematically independent events.\nc) Based on the results and comparing $P(A|S)$ with $P(A)$, interpret the effectiveness of the course for the Directorate-General for Traffic: would you advise recommending this course to future drivers? Justify your answer in terms of probability.

In an investigation into the cultural consumption of young people in Spain, two subscription services are analysed: general digital press ($D$) and on-demand music platforms ($M$). It is known that the probability that a young person subscribes to the digital press is $P(D) = 0,40$ and that the probability that they subscribe to at least one of the two services is $P(D \cup M) = 0,70$.\n\na) Set up an equation based on the properties of probability and deduce what the exact value of $P(M)$ must be for being subscribed to music and to the digital press to be statistically independent events.\nb) Suppose that a subsequent, broader survey reveals that the probability of subscribing to music is actually $P(M) = 0,60$. With this new data, calculate the actual value of $P(D \cap M)$ and determine whether the two services behave as events that «attract» each other (occur together more frequently than expected by pure chance) or «repel» each other (occur together less frequently than expected).

In a popular strategy video game, players open "legendary chests" to obtain rewards. The probability of getting an S-level hero from the first chest is only $0,05$. However, the game has an "accumulated luck" system: if you do not get the hero from a chest, the probability of getting it from the next one increases by $0,02$ compared with the probability for the previous chest. If you get the hero, the probability goes back to $0,05$ for the next attempt.

The figure shows the situation you must study: the first two chests do not give the hero and the hero appears in the third.

Calculate the probability that a player needs to open exactly three chests to get their first S-level hero.

An urn contains $n$ white balls and $4$ black balls. Two balls are drawn successively without replacement. Knowing that the probability that both balls are white is exactly $\frac{1}{3}$, determine using an algebraic equation how many white balls were initially in the urn.

A two-factor authentication security system sends a code to the user's mobile phone. If the first message does not arrive (which occurs with a probability of $0,12$), the system attempts a second send. However, due to network congestion, if the first send failed, the probability of the second also failing increases to $0,45$. On the other hand, if the first send was successful, the probability of the user receiving a second confirmation message is $0,98$.

Calculate the probability that a user receives the second message, regardless of what happened with the first.

In basketball practice, a player shoots two consecutive free throws. The probability of scoring the first is $0,7$. If he scores the first, his confidence increases and the probability of scoring the second is $0,8$. If he misses the first, he gets nervous and the probability of scoring the second drops to $0,4$.

If we know that the player scored only one of the two shots, what is the probability that it was the second?

Three pharmaceutical laboratories ($L_1$, $L_2$ and $L_3$) manufacture the same type of antigen test. $L_1$ produces $45\%$ of the total, $L_2$ produces $30\%$ and $L_3$ produces the rest. It is known that the percentage of defective tests is $1\%$ for $L_1$, $2\%$ for $L_2$ and $3\%$ for $L_3$.

If we choose two tests at random from a large batch where all manufacturers are mixed together (you may consider the drawing to be with replacement due to the large volume):

1. a) What is the probability that both are defective?
2. b) Given that the first turned out to be defective, what is the probability that the second is also defective? Justify whether these events are dependent or independent.

Two events $A$ and $B$ satisfy $P(A) = 0,6$ and $P(B) = 0,7$. The figure shows a Venn diagram with both events inside the sample space $\Omega$.

1. a) Determine the interval of possible values for the probability of the intersection $P(A \cap B)$.

1. b) Based on this, explain why the events $A$ and $B$ can never be mutually exclusive.

1. c) What value should $P(A \cap B)$ have for $A$ and $B$ to be independent?

At an intelligence centre, facial recognition software analyzes images. The probability that it correctly identifies an individual if they are in the database is $0,95$. However, the software has a margin of error and sometimes incorrectly identifies people who are not in the database with a probability of $0,02$.

Given that only $1\%$ of the people passing through customs are in the database, calculate:

1. a) The probability that the software triggers an alert (identifies someone).
2. b) If the software has identified a person, what is the probability that they are actually in the database?

Two tetrahedral dice (faces numbered 1 to 4) are rolled. Let the events be:

• $A$: "The sum of the faces is a prime number".

• $B$: "The product of the faces is a multiple of 4".

• $C$: "At least one of the dice shows a 1".

Study the independence of events $A$ and $B$, and of events $B$ and $C$. Is there any dependence relationship between the outcome of the first die and the outcome of the second? Explain mathematically.

A critical data storage server at a bank has two independent backup systems, $S_1$ and $S_2$. In the event of accidental file loss, the probability that system $S_1$ recovers the data is $0,90$, and that of system $S_2$ is $0,80$. To calculate the probability that the bank does not lose the data (that is, that at least one of the two systems works successfully, event $S_1 \cup S_2$), two analysts propose different procedures:\n- Method 1: Apply the general addition rule $P(S_1 \cup S_2) = P(S_1) + P(S_2) - P(S_1 \cap S_2)$, using the independence of $S_1$ and $S_2$ to calculate the intersection.\n- Method 2: Use the complementary event and calculate $1 - P(\overline{S_1} \cap \overline{S_2})$, assuming that if two events are independent, their complements are also independent.\n\na) Work out both methods numerically and verify that they yield the identical result.\nb) Prove in a general algebraic manner that if any two events $A$ and $B$ are independent, then their complementary events $\overline{A}$ and $\overline{B}$ are necessarily independent.\nc) If the bank decides to increase security by installing $5$ independent backup systems instead of $2$, justify which of the two methods is noticeably more efficient and advantageous for calculating the overall probability of success.

A pseudorandom security generator for bank transfers outputs two-bit codes with four equally likely outcomes: $\Omega = \{00, 01, 10, 11\}$, each with probability $\frac{1}{4}$. We consider the following three events:\n- $A$: «the first bit is $1$»\n- $B$: «the second bit is $1$»\n- $C$: «exactly one of the two bits is $1$» (that is, the bits are distinct).\n\na) Describe the elements that make up events $A$, $B$, and $C$, and calculate their probabilities $P(A)$, $P(B)$, and $P(C)$.\nb) Analyse pairwise independence: mathematically check whether $A$ and $B$ are independent, whether $A$ and $C$ are independent, and whether $B$ and $C$ are independent.\nc) Check whether the three events are mutually independent by evaluating the equality $P(A \cap B \cap C) = P(A) \cdot P(B) \cdot P(C)$.\nd) Explain the apparent paradox: how is it possible that knowing the outcome of one event individually provides no information about another (pairwise independence), yet knowing the joint outcome of two of them determines with $100\,\%$ certainty what happens with the third?

Topic challenge. Does educational attainment determine youth labour market integration? Analyse official data from the INE to test this probabilistically.

1. Search and counting: Access the INE portal (EPA) and extract data for a $2 \times 2$ contingency table on young people (aged 16-24), classifying them according to their educational level (Higher education vs. Non-higher education) and employment status (Employed vs. Unemployed).

2. Spreadsheet processing: Enter the values into a spreadsheet. Define the events $S$ ("holds higher education qualifications") and $D$ ("is unemployed"). Assign marginal and compound probabilities using Laplace's rule and matrix counting techniques.

3. Conditional probability and independence: Calculate $P(D|S)$ and $P(D|\overline{S})$ in the spreadsheet. Formally justify using theoretical properties whether $S$ and $D$ are dependent or independent by checking if $P(S \cap D) = P(S) \cdot P(D)$ with a tolerance margin of less than $0,005$.

4. Reasoned proposal: Write a one-page executive report with your tables, formulas used, and a sociological assessment strictly based on the probabilistic results obtained.

Assessment criteria:

• Rigorous collection of real data and correct mathematical formulation in the spreadsheet.

• Precision in calculating compound and conditional probabilities, and in demonstrating (in)dependence.

• Critical ability to argue socioeconomic conclusions based on probabilistic values.