Exercises: Conditional probability

A survey on the use of glasses and the gender of the students has been carried out in a high school senior class. The results are shown in the following contingency table:

If we select a student at random, calculate:

1. a) The probability that they are a boy, given that they wear glasses.
2. b) The probability that they wear glasses, given that they are a girl.

On a streaming content platform, it is known that the probability of a user watching a science fiction series is $P(S) = 0,6$ and the probability of them watching a documentary is $P(D) = 0,4$. Furthermore, the probability of them watching both types of content is $P(S \cap D) = 0,15$.

Calculate the probability that a user watches a documentary if we know they have already watched a science fiction series.

A social media analyst studies the behavior of teenagers on two applications: TikTok ($T$) and Instagram ($I$). The data indicates that:

• $70 \%$ of young people use TikTok.

• $50 \%$ use Instagram.

• The probability that a young person uses Instagram given that they use TikTok is $0,4$.

Calculate the probability that a randomly chosen young person uses both applications ($P(T \cap I)$) and determine whether the events are independent.

In a pencil case there are 5 highlighters: 3 are yellow and 2 are pink. We draw two highlighters consecutively and without replacement (without returning the first one to the pencil case).

Represent the situation using a tree diagram and calculate the probability that the second highlighter drawn is pink, given that the first was yellow.

Let $A$ and $B$ be two events of a random experiment such that the probability of $A$ is $P(A) = 0,5$, that of $B$ is $P(B) = 0,3$ and the probability of their union is $P(A \cup B) = 0,65$.

From the following Venn diagram, identify the shaded area and calculate $P(A|B)$.

In a video game tournament, $40 \%$ of the participants are experts ($E$) and the rest are beginners ($P$). It is known that $80 \%$ of the experts reach the final ($F$), whereas only $20 \%$ of the beginners achieve this.

If we choose a participant at random and we know they are a beginner, what is the probability that they reach the final? Express the result using conditional probability notation.

A study has been carried out on the reading habits of a group of students. The probability that a student reads digital news is $P(L) = 0,45$. The probability that they read digital news and also listen to news podcasts is $P(L \cap S) = 0,18$.

Find the probability that a student listens to news podcasts, given that they are a reader of digital news.

In a sportswear store, $60 \%$ of sales are running shoes ($Z$) and $40 \%$ are basketball shoes ($B$). $20 \%$ of the buyers of running shoes also purchase technical socks ($C$).

If a customer has bought running shoes, what is the probability that they have not bought technical socks? (Remember the property of the complementary event in conditional probability).

In a survey of 200 high school students about the use of social networks, it was observed that 120 use TikTok, 100 use Instagram, and 60 use both platforms. If we select a student at random, calculate:

1. a) The probability that they use Instagram given that they already use TikTok.
2. b) The probability that they do not use TikTok if we know they use Instagram.
3. c) Are the events "being a TikTok user" and "being an Instagram user" independent? Justify your answer mathematically.

An urban clothing brand manufactures its hoodies in two different workshops. Workshop $A$ produces $60\%$ of the garments and workshop $B$ produces the rest. It is known that $5\%$ of the hoodies from workshop $A$ have a stitching defect, whereas in workshop $B$ the percentage of defects is $8\%$. A hoodie is chosen at random from the store and turns out to be defective. Calculate the probability that it was manufactured in workshop $A$.

A high school senior has prepared 18 of the 25 topics that make up the final History exam. The exam consists of choosing two topics at random from the total and writing about one of them.

1. a) Calculate the probability that the student knows both drawn topics.
2. b) Calculate the probability that the student passes the exam (that is, they know at least one of the two topics).

On a streaming platform, $70\%$ of subscribers watch action series and $45\%$ watch documentaries. Furthermore, it is known that $30\%$ of users watch both types of content. If we select a user who we know does not watch documentaries, what is the probability that they do watch action series?

A sports training center classifies its athletes according to their diet (Balanced or Deficient) and their performance in the last quarter (High or Low). The data is collected in the following contingency table:

Calculate:

1. a) $P(R | D)$
2. b) $P(\bar{D} | \bar{R})$
3. c) Determine whether having a balanced diet and having a high performance are independent events.

A screening test for a rare disease, which $1\%$ of the population suffers from, tests positive in $96\%$ of sick people (sensitivity). However, it also gives a "false positive" in $2\%$ of healthy people.

1. a) Calculate the probability that a randomly chosen person tests positive.
2. b) If a person's test has come back positive, what is the actual probability that they are sick?

Let $A$ and $B$ be two events of a random experiment such that $P(A) = 0.4$, $P(B) = 0.6$ and $P(A \cup B) = 0.8$.

1. a) Calculate $P(A | B)$.
2. b) Calculate $P(B | \bar{A})$.
3. c) Are $A$ and $B$ mutually exclusive events? Are they independent?

In a cooperative video game, player 1 has a $0.8$ probability of completing their mission. Player 2, independently of the first, has a $0.7$ probability of success.

1. a) What is the probability that both complete their missions?
2. b) Given that at least one of the two has completed their mission, what is the probability that player 1 has completed it?

On a streaming platform, $60 \%$ of users have a "Standard" subscription ($S_1$) and the rest have a "Premium" subscription ($S_2$). It is known that $15 \%$ of users with a standard subscription cancel their account before the end of the first month, whereas among Premium users, only $5 \%$ cancel in that period.

If we select a random user who has canceled their subscription before the first month, calculate the probability that they had a "Premium" subscription. Illustrate the problem using a tree diagram showing all the probabilities involved.

We have two opaque urns. Urn $U_A$ contains 4 white balls and 6 black balls. Urn $U_B$ contains 5 white balls and 3 black balls. The experiment consists of drawing a ball from urn $U_A$ and, without looking at its color, placing it into urn $U_B$. Next, two balls are drawn simultaneously from urn $U_B$.

If the two balls drawn from the second urn turn out to be white, what is the probability that the ball transferred from the first urn was black? Express the result as a fraction in lowest terms.

A study on digital habits in teenagers reveals that $70 \%$ of students at a school use a microblogging social network ($M$). It is known that $40 \%$ of the users of this network have experienced digital fatigue ($F$), whereas among those who do not use it, the percentage of digital fatigue drops to $10 \%$.

Calculate the following probabilities:

1. The probability that a randomly chosen student does not use the social network and does not have digital fatigue.

2. If a student claims to have digital fatigue, what is the probability that they are not a user of this social network?

3. Are the events "being a user of the social network" and "having digital fatigue" independent? Justify your answer mathematically.

A factory making components for smartphones uses three machines, $A$, $B$, and $C$, to produce a specific part. Machine $A$ produces $45 \%$ of the total, $B$ produces $30 \%$, and $C$ produces the rest. It has been detected that $2 \%$ of the parts from $A$ are defective, $3 \%$ of those from $B$, and $5 \%$ of those from $C$.

If a part is chosen at random and turns out to be defective, determine with reasoning which machine is most likely to have manufactured it. To do this, calculate the three a posteriori probabilities using Bayes' Theorem.

Let $A$ and $B$ be two events in a sample space such that we know $P(A \cup B) = 0,9$, $P(B|A) = 0,4$ and $P(A|\overline{B}) = 0,8$, where $\overline{B}$ represents the complementary event of $B$.

Using the properties of the algebra of events and the definition of conditional probability, calculate:

1. $P(A \cap B)$

2. $P(A)$

3. $P(B)$

Hint: Set up a system of equations using the definitions of conditional probability.

In an esports tournament, a player must pass two consecutive stages. The probability of passing the first stage is $0,7$. If they pass the first, the probability of passing the second is $0,6$. However, if they do not pass the first stage, they have a "second chance" opportunity where the probability of passing the second stage drops to $0,2$.

1. Calculate the probability that the player passes the second stage.

2. Given that the player has passed the second stage, what is the probability that they did NOT originally pass the first stage?

A health center conducts a screening test for a food intolerance that affects $2 \%$ of the population. The test is not perfect: it tests positive in $98 \%$ of the people who actually suffer from the intolerance, but it also gives a "false positive" in $3 \%$ of healthy people.

If a teenager takes the test and the result is positive:

1. What is the probability that they are actually intolerant?

2. Interpret the result obtained in relation to the effectiveness of the test for mass screening in populations where the prevalence of the disease is very low.

In a high school senior classroom, $60 \%$ of the students are female. It is known that $80 \%$ of the females passed the last math exam, whereas the percentage of passes among the males was $x \%$.

If the probability that a randomly chosen person from the classroom has passed the exam is $0,74$, calculate the value of $x$ (percentage of passes among males). Then, calculate the probability that a randomly chosen person is male, given that they have failed the exam.