Exercises: Dependent and independent events

In a random experiment, a fair coin (with sides $C$ and $X$) and a six-sided die numbered 1 to 6 are rolled. Consider the following events:

• $A$: "Getting heads on the coin".

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

Reason whether the 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 one after the other with replacement (the first ball is returned to the urn before the second one is drawn).

1. a) Calculate the probability that both balls drawn are orange.

1. b) Does the result of the first draw affect the probability of the second? Justify your answer.

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

1. a) Calculate 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 on the first draw.

Let $A$ and $B$ be two events in a sample space such that their probabilities are known: $P(A) = 0.6$ 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 2nd Baccalaureate class, $40\%$ of 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, calculate the probability that:

1. a) They use both social networks.
2. b) They use neither of them.

A basketball player has a probability of $0.8$ of making a free throw. If they shoot two free throws in a row and we assume that success in the second shot is independent of the outcome 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 shot.

Two cards are drawn successively and without replacement from a Spanish deck of 40 cards (there are 4 Kings). Let the events be:

• $A$: "The first card is a King".

• $B$: "The second card is a King".

The tree diagram shows the situation. The probabilities of the first draw are already written; the ones for the second draw, marked with «?», you have to determine yourself.

Calculate the probability of $B$ conditioned on $A$, that is, $P(B|A)$. Is this probability equal to the probability of getting a King on 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 the events $X$ and $Y$ are independent. To do so, check whether the necessary condition relating the probability of the intersection to the product of the individual probabilities is satisfied.

In a study on streaming content consumption among young people, it was observed that the probability that a student is subscribed to platform A is $P(A) = 0.6$, whereas the probability that they are subscribed to platform B is $P(B) = 0.5$. Furthermore, it is known that the probability of not being subscribed to either platform 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 $P(A) \times P(B)$.

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

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

In a school, $45\%$ of the 2nd-year Bachillerato 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 that represents this situation, indicating the probability on each branch.

1. b) Is liking statistics an event independent of the Bachillerato 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: one for video games ($V$) and one for cooking ($C$). $60\%$ of their subscribers follow channel $V$, $40\%$ follow channel $C$ and $20\%$ follow both channels.

1. a) If we randomly select a subscriber who is known to follow the cooking channel, what is the probability that they also follow the video games channel?
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 felt-tip pens: 6 are blue and 4 are red. Two felt-tip pens are drawn successively without replacement.

1. a) What is the probability that both are blue?
2. b) What is the probability that the second is red given that the first was blue?
3. c) Are the events "the first felt-tip pen is blue" and "the second felt-tip pen 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 are shown 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) Given 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 reasons whether the events "being under 25 years old" and "driving a moped" are independent.

Consider two events $A$ and $B$ of a random experiment with the following known probabilities: $P(A) = 0.5$, $P(B | A) = 0.4$ 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) Show whether the events $A$ and $B$ are independent. What conclusion can you draw about the probability of $B$ when it does not change whether or not we know that $A$ has 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 $0.45$ and that it is recommended for its popularity is $0.4$. Knowing that a user likes a series, the probability that the algorithm recommended it for its genre is $0.5$. Furthermore, the events "liking the series" and "being recommended for its popularity" are independent.

Calculate, showing your reasoning:

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 your answer with calculations.

We have two urns with the following contents: 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 $U_2$ turn 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 of node $N_1$ failing is $0.02$. Node $N_2$ is more sensitive: if $N_1$ has validated the entry, the probability of $N_2$ failing is $0.05$, but if $N_1$ has failed, node $N_2$ locks up and fails with probability $1$. Finally, node $N_3$ operates independently of the others 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 energy, 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 $0.8$, whereas for a passive individual this probability drops 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 (affecting $0.5\%$ of the population) gives $2\%$ false positives (healthy people who test positive) and $1\%$ false negatives (sick people who test negative).

1. a) If a person takes the test and tests positive, and immediately afterwards 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 school organises lunch into three sittings. $20\%$ of students eat in the first sitting, $50\%$ in the second and $30\%$ in the third. Punctuality when returning to class depends on the sitting: from the first sitting $10\%$ arrive late, from the second $5\%$ and from the third $x\%$. Given that the probability that a randomly chosen student arrives on time for class is $0.923$:

1. a) Determine the value of $x$ (percentage of students who arrive late from the third sitting).
2. b) If a student has arrived late for class, from which sitting are they most likely to be?

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 playoff ("repesca") opportunity which they pass with probability $0.3$. If they pass the playoff, 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 ultimately passes the tournament (passes the second stage).
2. b) If the player has passed the tournament, what is the probability that they did so without needing to go through the playoff?