Exercises: Conditional probability

A card is drawn at random from a 40-card Spanish deck. We consider the events $A$: "drawing a king" and $B$: "drawing a face card" (jack, knight, or king). Calculate the probability that the card is a king knowing that a face card has been drawn, that is, determine $P(A|B)$.

A survey was conducted among 100 high school juniors about their preference between two social media platforms. The data is collected in the following contingency table:

If we choose a student at random and they turn out to be a girl, what is the probability that she prefers TikTok? Calculate $P(T|M)$.

Given two events $A$ and $B$ of a random experiment, it is known that the probability of their intersection is $P(A \cap B) = 0,12$ and that the probability of event $B$ is $P(B) = 0,4$. Calculate the probability that $A$ occurs knowing that $B$ has occurred.

An urn contains 5 blue balls and 3 red balls. We draw two balls consecutively with replacement (we return the first ball to the urn before drawing the second).

1. a) Represent the experiment using a tree diagram.
2. b) Calculate the probability that the second ball is red knowing that the first was blue. Are they dependent or independent events?

In a group of friends, $60\%$ use Spotify ($S$), $40\%$ use Netflix ($N$) and $20\%$ use both platforms. If we choose a person who we know uses Netflix, what is the probability that they also use Spotify?

A professional darts player hits the bullseye $80\%$ of the time. If they make two independent throws, calculate the probability that:

1. a) They hit the second throw knowing that they hit the first.
2. b) They hit both throws in a row.

In a language center, $30\%$ of the students study French ($F$). Of these, $60\%$ are high school students ($B$). If we select a student at random from the center's list, what is the probability that they study French and are a high school student? Use the conditional probability formula: $P(F \cap B) = P(F) \cdot P(B|F)$.

We draw two cards from a Spanish deck without replacement. Calculate the probability that the second card is an ace, knowing that the first card drawn was the ace of coins.

A study has been carried out in a high school in Chicago on the leisure habits of $100$ 11th-grade students. The data has been collected in the following contingency table, classifying the students by their gender and their preferred social media platform (Instagram or TikTok):

If we choose a student at random, calculate the following probabilities using conditional probability notation:

1. a) That they prefer TikTok, knowing that they are a girl.
2. b) That they are a boy, given that their preferred social media platform is Instagram.
3. c) Are the events "being a boy" ($H$) and "preferring Instagram" ($I$) independent? Justify your answer by comparing $P(I/H)$ with $P(I)$.

A toolbox contains $12$ LED bulbs, of which $4$ are defective. If we draw two bulbs from the box consecutively and without replacement, calculate the probability of the following events:

1. a) That the second bulb is defective, knowing that the first one drawn worked correctly.
2. b) That both bulbs are defective.
3. c) That at least one of the two bulbs works correctly.

In an Esports league, a team has a $70\%$ probability of winning the match if they play on their local server ($L$), but this probability drops to $40\%$ if they play on an external server ($E$). It is known that they play $60\%$ of their matches locally.

1. a) Represent the situation using a tree diagram.
2. b) If the team has won their last match, what is the probability that they played on their home server?
3. c) What is the total probability that the team wins any given match?

Let $A$ and $B$ be two events of a random experiment such that $P(A) = 0,5$, $P(B) = 0,7$ and $P(A \cup B) = 0,85$.

1. a) Calculate $P(A \cap B)$.
2. b) Find the probability that $A$ occurs knowing that $B$ has occurred.
3. c) Determine whether events $A$ and $B$ are compatible and whether they are independent.

A medical study indicates that the probability of a person from a certain population having a spring allergy is $0,15$. A test has been developed to detect it, but it is not perfect:

• If a person has an allergy, the test is positive in $95\%$ of cases.

• If a person does not have an allergy, the test is positive (false positive) in $4\%$ of cases.

If a person is chosen at random and the test is positive, what is the probability that they actually have an allergy?

In a language academy, $40\%$ of the students study French ($F$), $30\%$ study German ($G$) and the rest study Italian ($I$). It is known that the percentage of students who pass the final exam is $60\%$ in French, $50\%$ in German and $80\%$ in Italian.

1. a) If we choose a student who has passed the exam, what is the probability that they study German?
2. b) If we choose a student at random, what is the probability that they have not passed the exam?

Two fair six-sided dice with faces numbered from $1$ to $6$ are rolled. We consider the following events:

• $S$: "The sum of the numbers rolled is equal to $7$".

• $A$: "An even number has been rolled on the first die".

Calculate:

1. a) The sample space of the sum of the numbers rolled and the probability of event $S$.
2. b) The probability that the sum is $7$ knowing that an even number has been rolled on the first die, $P(S/A)$.
3. c) Does the fact that the first die is even influence the probability that the sum is $7$? Justify whether the events are dependent or independent.

On a streaming platform, $20\%$ of users prefer horror movies, $50\%$ prefer comedies and $30\%$ prefer dramas. The data analysis department has observed that the probability of a user finishing watching a full movie depends on the genre:

• Horror: $0,6$

• Comedy: $0,8$

• Drama: $0,7$

If we select a user at random and see that they have left a movie halfway through (they have not finished it), what is the probability that the movie was a horror movie?

Let $A$ and $B$ be two events of the same random experiment such that the following probabilities are known:

• $P(A^C \cup B^C) = 0,8$

• $P(A^C | B) = 0,5$

• $P(A \cup B) = 0,75$

Calculate, justifying each step using the properties of probability:

1. a) $P(A \cap B)$
2. b) $P(B)$
3. c) $P(A)$
4. d) Determine with reasoning whether events $A$ and $B$ are independent.

In a cybersecurity laboratory, an artificial intelligence algorithm analyses emails to detect phishing. It is known that:

• $5\%$ of the emails the company receives are scam attempts (phishing).

• If an email is phishing, the algorithm correctly identifies it as "suspicious" in $98\%$ of cases.

• If an email is legitimate, the algorithm makes a mistake and marks it as "suspicious" in $3\%$ of occasions (false positive).

If an employee receives a "suspicious email" alert in their inbox:

1. a) What is the real probability that the email is actually a phishing attempt?
2. b) If the algorithm indicates that the email is "safe", what is the probability that it is a mistake and is actually phishing?

We have two urns with the following composition:

• Urn A: 4 red balls and 6 blue balls.

• Urn B: 7 red balls and 3 blue balls.

The experiment consists of rolling a fair six-sided die. If a prime number is rolled, we draw a ball from Urn A and place it in Urn B without looking at its color; then, we draw a ball from Urn B. If a non-prime number is rolled, we directly draw a ball from Urn B.

Calculate the probability that the final ball drawn from Urn B is red.

A streaming platform uses three servers ($S_1, S_2, S_3$) to distribute its content. $40\%$ of the traffic is managed from $S_1$, $35\%$ from $S_2$ and the rest from $S_3$. Due to network load, micro-cuts in playback occur with the following probabilities:

• On server $S_1$, the probability of a micro-cut is $2\%$.

• On server $S_2$, the probability of a micro-cut is $5\%$.

• On server $S_3$, the probability of a micro-cut is $8\%$.

1. a) If a user is watching a show, what is the probability that they experience a micro-cut?
2. b) If a user calls technical support reporting a micro-cut, what is the probability that their connection is being served by server $S_3$?

In an Esports competition, a League of Legends player has a $70\%$ probability of winning if they choose their favorite character (Main). If they do not choose it, their probability of winning drops to $45\%$. The matchmaking system allows the player to get their favorite character $60\%$ of the time.

1. a) Calculate the probability that the player wins their next match.
2. b) Knowing that the player has lost the match, what is the probability that they were not able to play with their favorite character?
3. c) Are the events "winning the match" and "playing with the favorite character" independent? Justify your answer by comparing probabilities.

A study is carried out on the use of social media in a high school in Chicago. The data shows that:

• $80\%$ of the students use Instagram ($I$).

• $60\%$ of the students use TikTok ($T$).

• $30\%$ of those who use Instagram do not use TikTok.

We choose a student at random. Calculate:

1. a) The probability that they use both social media platforms.
2. b) Knowing that they use TikTok, the probability that they also use Instagram.
3. c) The probability that they use neither of the two networks.

A logistics company has two routes to deliver orders in the city center. Route A is used $70\%$ of the time and route B the remaining $30\%$.

• Due to traffic, on route A the delivery driver arrives late $15\%$ of the days.

• On route B, the delivery driver arrives late $25\%$ of the days.

On a given day, the manager observes that an order has arrived late. The delivery driver states: "If I had taken route A, the probability of arriving on time would have been greater than the probability of having come today via route A given that I arrived late". Prove mathematically whether the delivery driver's statement is true or false by calculating both probabilities.

In a multiple-choice exam, a student knows the correct answer to a question with probability $p$ and guesses it at random with probability $1-p$. Each question has $k$ answer options, of which only one is correct.

1. a) Show that the probability that the student answers the question correctly is $\frac{1 + (k-1)p}{k}$.
2. b) Knowing that the student has answered correctly, what is the probability that they actually knew the answer and did not guess it at random? Express the result in terms of $p$ and $k$.
3. c) If a question has 4 options ($k=4$) and the student has a $50\%$ probability of knowing the answer ($p=0,5$), calculate the numerical value of the previous section and interpret it.