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 colors: 10 red, 6 blue and 4 green. A child draws a gumball, notes its color, 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 high 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 soccer.
2. b) $P(F|H)$, the probability that they like soccer given that they are a boy.
3. c) In light of the results, are the events "being a boy" and "liking soccer" independent?

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 color.

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 high school student has two independent alarm clocks on their cell 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 $B$.

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 cell 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 center, 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.