Exercises: Dependent and independent events

In a random experiment, a fair coin (with faces $C$ and $X$) and a six-sided die numbered from 1 to 6 are flipped and rolled. The following events are considered:

• $A$: "Obtaining heads on the coin".

• $B$: "Obtaining an even number on the die".

Reason whether events $A$ and $B$ are dependent or independent and calculate the probability that both occur at the same time, that is, $P(A \cap B)$.

An urn contains 5 orange balls and 3 purple balls. Two balls are drawn consecutively with replacement (the first ball is returned to the urn before drawing the second).

Calculate the probability that the two drawn balls are orange. Does the result of the first draw influence the probability of the second?

Using the same urn from the previous exercise (5 orange balls and 3 purple balls), two balls are now drawn without replacement.

1. a) Find the probability that the second ball is orange given that the first was purple.
2. b) Are these events independent? Justify your answer by comparing this result with the probability of drawing an orange ball in the first position.

Let $A$ and $B$ be two events in a sample space such that we know their probabilities: $P(A) = 0,6$ and $P(B) = 0,4$. If we know that the events are independent, calculate:

1. a) $P(A \cap B)$
2. b) $P(A \cup B)$
3. c) $P(A|B)$

In a high school senior class, $40\%$ of the students regularly use the social network Instagram ($I$) and $25\%$ use TikTok ($T$). It is also known that the use of both networks is independent of each other. If we choose a student at random, find the probability that they:

1. a) Use both social networks.
2. b) Use neither of the two.

A basketball player has a probability of making a free throw of $0,8$. If they shoot two consecutive free throws and we assume that success in the second shot is independent of the result of the first, calculate:

1. a) The probability that they make both shots.
2. b) The probability that they miss both shots.
3. c) The probability that they make at least one.

Two cards are drawn successively and without replacement from a 40-card Spanish deck. Let the events be:

• $A$: "The first card is a King".

• $B$: "The second card is a King".

Calculate the probability of $B$ conditional on $A$, that is, $P(B|A)$. Is this probability equal to the probability of obtaining a King in the first draw? Justify whether the events are dependent or independent.

The following information is available about two events $X$ and $Y$:

$P(X) = 0,5$, $P(Y) = 0,3$ and $P(X \cap Y) = 0,15$.

Prove mathematically whether events $X$ and $Y$ are independent. To do this, check if the necessary condition relating the probability of the intersection to the product of the individual probabilities is met.

In a study on the consumption of streaming content among young people, it has been observed that the probability that a student is subscribed to platform A is $P(A) = 0,6$, while the probability that they are subscribed to platform B is $P(B) = 0,5$. Furthermore, it is known that the probability that they are not subscribed to either of the two is $P(\bar{A} \cap \bar{B}) = 0,2$.

1. a) Calculate the probability that a student is subscribed to both platforms.
2. b) Determine whether the events "being subscribed to platform A" and "being subscribed to platform B" are independent events. Justify your answer by comparing $P(A \cap B)$ with $P(A) \cdot P(B)$.

A sportswear brand knows that $70\%$ of its customers buy running shoes. Of those who buy running shoes, $40\%$ also purchase an athletic t-shirt. On the other hand, among those who do not buy running shoes, only $15\%$ purchase the athletic t-shirt.

1. a) If we choose a customer at random, what is the probability that they buy the athletic t-shirt?
2. b) Are the events "buying running shoes" and "buying a t-shirt" independent? Reason your answer by analyzing whether the fact of buying running shoes modifies the probability of buying the t-shirt.

In a high school, $45\%$ of the senior class students are in the Social Sciences track ($CS$), $35\%$ in Science and Technology ($CT$), and the rest in Arts ($A$). It is known that the probability that a student likes statistics is $0,8$ if they are in $CS$, $0,6$ if they are in $CT$, and $0,3$ if they are in $A$.

1. a) Draw the tree diagram representing this situation.
2. b) Is liking statistics an event independent of the high school track studied? Prove it mathematically for the case of the Social Sciences track.

Let $A$ and $B$ be two events such that $P(A) = 0,4$ and $P(B) = 0,3$.

1. a) If $A$ and $B$ are independent events, calculate $P(A \cup B)$ and $P(A | B)$.
2. b) If $A$ and $B$ are dependent events and we know that $P(A \cap B) = 0,2$, calculate $P(B | A)$.
3. c) In case b), are $A$ and $B$ mutually exclusive events? Justify your answer.

A digital content creator has two channels: a gaming one ($V$) and a cooking one ($C$). $60\%$ of their subscribers follow channel $V$, $40\%$ follow channel $C$, and $20\%$ follow both channels.

1. a) If we select a subscriber at random who we know follows the cooking channel, what is the probability that they also follow the gaming one?
2. b) In light of the data, would you say that following one channel influences the probability of following the other? Justify whether the events are dependent or independent.

In a pencil case there are 10 markers: 6 are blue and 4 are red. Two markers are drawn successively and without replacement.

1. a) What is the probability that both are blue?
2. b) What is the probability that the second is red if the first was blue?
3. c) Are the events "the first marker is blue" and "the second marker is blue" independent? Justify your answer by comparing the conditional probabilities.

Traffic accidents in a city have been classified according to the type of vehicle and whether the driver was under 25 years old. The data is collected in the following table:

	Car ($T$)	Moped ($M$)	TOTAL
Under 25 years old ($J$)	$12$	$18$	$30$
Over 25 years old ($V$)	$48$	$22$	$70$
TOTAL	$60$	$40$	$100$

1. a) If we choose an accident at random, what is the probability that the driver is under 25 years old?
2. b) If we know that the accident involved a moped, what is the probability that the driver is under 25 years old?
3. c) Using the previous results, explain with reasoning whether the events "being under 25 years old" and "driving a moped" are independent.

Consider two events $A$ and $B$ of a random experiment for which the following probabilities are known: $P(A) = 0,5$, $P(B | A) = 0,4$ and $P(B | \bar{A}) = 0,4$.

1. a) Calculate $P(A \cap B)$ and $P(\bar{A} \cap B)$.
2. b) From the previous results, calculate $P(B)$.
3. c) Prove whether events $A$ and $B$ are independent. What conclusion can you draw about the probability of $B$ when it does not vary even if we know that $A$ has occurred or has not occurred?

On a streaming platform, the algorithm recommends a series based on two criteria: genre (event $G$) and popularity (event $P$). It is known that the probability that a user likes a series is $0,6$. The probability that it is recommended for its genre is $0,45$. Given that a user likes a series, the probability that the algorithm recommended it for its genre is $0,8$. Furthermore, the events "liking the series" and "being recommended for its popularity" are independent.

Calculate, justifying your steps:

1. a) The probability that a series is recommended for its genre and the user also likes it.
2. b) The probability that a user likes a series or it has been recommended for its popularity.
3. c) Are the events "being recommended for its genre" and "liking the series" independent? Justify the answer with calculations.

We have two urns with the following compositions: urn $U_1$ contains 4 red balls and 6 black balls; urn $U_2$ contains 3 red balls and 2 black balls. The experiment consists of drawing a ball at random from $U_1$ and placing it into $U_2$, without looking at its color. Next, two balls are drawn simultaneously from $U_2$.

1. a) Calculate the probability that the two balls drawn from $U_2$ are of different colors.
2. b) If the two balls drawn from $U_2$ turned out to be black, what is the probability that the ball transferred from $U_1$ was red?

An industrial cybersecurity system uses three verification nodes ($N_1, N_2, N_3$) arranged in series, so that the system only allows access if all three nodes validate the entry. The probability that node $N_1$ fails is $0,02$. Node $N_2$ is more sensitive: if $N_1$ has validated the entry, the probability that $N_2$ fails is $0,05$, but if $N_1$ has failed, node $N_2$ blocks and fails with probability $1$. Finally, node $N_3$ works independently of the rest with a success probability of $0,98$.

1. a) What is the probability that a legitimate entry is rejected by the system?
2. b) If the system has rejected an entry, calculate the probability that the failure occurred exclusively in node $N_2$.

In a sociological study on the use of renewable energies, individuals are classified as "proactive" ($A$) or "passive" ($B$). It is known that $30\%$ of the population is proactive. The probability that a proactive individual installs solar panels is $0,8$, while for a passive individual this probability is reduced to $0,2$. Two people are chosen at random independently.

1. a) Calculate the probability that at least one of the two people installs solar panels.
2. b) Given that only one of the two people has installed solar panels, what is the probability that this person belongs to the proactive group?

Let $A$ and $B$ be two events of a random experiment such that $P(A^c \cap B^c) = 0,2$, $P(A | B) = 0,4$ and $P(B | A) = 0,5$.

1. a) Calculate the probabilities of events $A$ and $B$. (Hint: use the definition of conditional probability to set up a system of equations).
2. b) Determine whether events $A$ and $B$ are independent. Justify your answer.
3. c) Calculate $P(A^c \cup B)$.

A diagnostic test for a rare disease (which affects $0,5\%$ of the population) yields $2\%$ of false positives (healthy people who test positive) and $1\%$ of false negatives (sick people who test negative).

1. a) If a person takes the test and tests positive, and immediately afterward takes a second identical and independent test which also tests positive, what is the probability that they are actually sick?
2. b) Comment on the result obtained by comparing it with the probability after a single positive test.

A courier company uses three routes for its deliveries: North, Central, and South. $20\%$ of the packages go via the North route, $50\%$ via the Central, and $30\%$ via the South. Delays depend on the route: on the North there are $10\%$ delays, on the Central $5\%$, and on the South $x\%$. Given that the probability that a package chosen at random arrives on time is $0,923$:

1. a) Determine the value of $x$ (percentage of delays on the South route).
2. b) If a package has arrived late, which route is most likely to have been the cause?

In a video game tournament, a player must pass two stages. The probability of passing the first stage is $0,7$. If they pass the first, the probability of passing the second is $0,5$. If they do not pass the first, they are given a "second chance" opportunity which they pass with probability $0,3$. If they pass the second chance, they enter the second stage, but with a success probability of only $0,2$ due to accumulated fatigue.

1. a) Calculate the probability that the player finally passes the tournament (passing the second stage).
2. b) If the player has passed the tournament, what is the probability that they achieved it without needing to go through the second chance?