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 it is known that the events are independent, calculate the probability that both occur 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 takes out a gumball, notes its color, puts it back into the machine and takes out a second gumball.

1. a) Justify whether the events "drawing a red ball first" and "drawing a blue ball 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 40-card Spanish deck, one after the other and without replacing the first one in the deck.

1. a) What is the probability that the first card is a King?
2. b) Knowing 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?

An 11th-grade student has two exams on the same day: Math and English. The probability of passing Math is $P(M) = 0,7$ and of passing English is $P(I) = 0,8$. Assuming that passing one exam does not influence the result of the other:

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

In a drawer in a clothing store fitting room, 4 black socks and 2 white socks were left behind, all of the same size and texture. If a store associate 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 the two socks drawn are black.

A survey has been 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 check at a light bulb factory, it is known that the probability of a light bulb being defective is $0,02$. 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 the extraction of one light bulb can be considered not to affect the probability of the next).

Observe 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 knowing that they are a boy.
3. c) In view 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 musical 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 balls drawn 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 one also being critical drops to $0,1$ due to the skill's cooldown. If the last hit was not critical, the probability of the next one remains at $0,2$.

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

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

In a high school in Chicago, a study was conducted on the use of social media platforms and the grade level of the students, obtaining the following data:

A student is chosen at random. Determine whether the event "preferring TikTok" is independent of the event "being in 11th grade."

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

1. a) Calculate the probability that the drone can continue flying if it is necessary for at least one of the two components to work correctly.

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

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

1. a) What is the probability that the second card is a King, knowing 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 alarms on their cell phone. The first ($D_1$) works $95\%$ of mornings and the second ($D_2$) works $90\%$ of the time.

1. a) Calculate the probability that the student oversleeps because neither of the two alarms goes off.

1. b) If we know that at least one of the alarms has gone off, what is the probability that alarm $D_1$ has 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 obtaining an S-tier hero in the first chest is only $0,05$. However, the game has an "accumulated luck" system: if you do not obtain the hero in a chest, the probability of obtaining it in the next one increases by $0,02$ with respect to the probability of the previous chest. If you obtain the hero, the probability returns to $0,05$ for the next attempt.

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

An urn contains $n$ white balls and $6$ black balls. Two balls are drawn successively without replacement. Knowing that the probability of both balls being 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 happens with a probability of $0,12$), the system attempts a second delivery. However, due to network congestion, if the first delivery failed, the probability of the second also failing increases to $0,45$. On the other hand, if the first delivery 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 one.

In a basketball training session, a player shoots two consecutive free throws. The probability of scoring the first is $0,7$. If they score the first, their confidence increases and the probability of scoring the second is $0,8$. If they miss the first, they get nervous and the probability of scoring the second drops to $0,4$.

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

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 (you can consider the extraction to be with replacement due to the large volume):

1. a) What is the probability that both are defective?
2. b) Knowing that the first one was defective, what is the probability that the second one is also defective? Justify whether these events are dependent or independent.

Two events $A$ and $B$ satisfy $P(A) = 0,6$ and $P(B) = 0,7$. Determine the interval of possible values for the probability of the intersection $P(A \cap B)$ and, based on this, reason why the events $A$ and $B$ can never be incompatible. What value should $P(A \cap B)$ have for $A$ and $B$ to be independent?

In 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 $0,02$.

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

1. a) The probability that the software gives 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 relationship of dependence between the result of the first die and the result of the second? Explain it mathematically.