Exercises: Conditional probability

In a Year 13 class, a survey was conducted on the use of glasses and the gender of students. The results are shown in the following contingency table:

If a student is chosen 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.

In a local library, it is known that the probability that a member borrows a novel is $P(N) = 0.6$ and the probability that they borrow a comic is $P(C) = 0.4$. Furthermore, the probability that they borrow both is $P(N \cap C) = 0.15$.

Calculate the probability that a member borrows a comic if we know that they have already borrowed a novel.

A social media analyst studies the behavior of teenagers on two apps: TikTok ($T$) and Instagram ($I$). The data indicate 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 apps ($P(T \cap I)$) and reason whether the events are independent.

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

Represent the situation with a tree diagram and calculate the probability that the second highlighter drawn is pink, given that the first one 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 region and calculate $P(A|B)$.

In a school basketball league, $40 \%$ of the participants are experienced players ($E$) and the rest are beginners ($P$). It is known that $80 \%$ of the experienced players reach the final ($F$), whereas only $20 \%$ of the beginners do so.

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

A study has been conducted on the reading habits of a group of students. The probability that a student reads digital press is $P(L) = 0.45$. The probability that they read digital press 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 read digital press.

In a sports clothing shop, $60 \%$ of sales are running shoes ($Z$) and $40 \%$ are basketball shoes ($B$). $20 \%$ of 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 complementary event property in conditional probability).

In a survey conducted on 200 students at a high school regarding social media use, it was observed that 120 use TikTok, 100 use Instagram, and 60 use both platforms. If a student is chosen 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 that they use Instagram.
3. c) Are the events "being a TikTok user" and "being an Instagram user" independent? Justify your answer mathematically.

A streetwear brand manufactures its hoodies in two different workshops. Workshop $A$ produces $60\%$ of the garments and workshop $B$ 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 shop and turns out to be defective. Calculate the probability that it was manufactured in workshop $A$.

A Year 13 student has prepared 18 out of the 25 topics included in the final History exam. The exam consists of choosing two topics at random from the total and answering one of them.

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

In a book club, $70\%$ of the members read mystery novels and $45\%$ read essays. Furthermore, it is known that $30\%$ of the members read both types of books. If we select a member who we know does not read essays, what is the probability that they do read mystery novels?

A sports performance center classifies its athletes according to their diet (Balanced or Deficient) and their performance in the last term (High or Low). The data are gathered 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 affects $1\%$ of the population, gives a positive result in $96\%$ of patients with the disease (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 result is positive, what is the actual probability that they have the disease?

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 track relay event, the first runner has a probability of $0.8$ of completing her leg without a penalty. The second runner, independently of the first, has a probability of $0.7$.

1. a) What is the probability that both complete their leg without a penalty?
2. b) Given that at least one of the two has completed her leg without a penalty, what is the probability that the first runner succeeded?

In a gym, $60 \%$ of members have a "Monthly" membership ($S_1$) and the rest an "Annual" membership ($S_2$). It is known that $15 \%$ of members with a monthly membership cancel before the end of the first month, whereas among members with an annual membership only $5 \%$ cancel in that period.

If we randomly select a member who has cancelled before the first month, calculate the probability that they had a monthly membership.

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 an irreducible fraction.

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

The following tree diagram shows these data on each branch:

Calculate the following probabilities:

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

1. b) If a student states that they have digital fatigue, what is the probability that they are not a user of that social network?

1. c) Are the events «being a user of the social network» and «having digital fatigue» independent? Justify your answer mathematically.

A smartphone components factory uses three machines, $A$, $B$ and $C$, to produce a specific part. Machine $A$ produces $45 \%$ of the total, $B$ produces $30 \%$ and $C$ the rest. It has been found 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 reasons which machine is most likely to have manufactured it. To do so, calculate the three posterior 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 the selection process for a school exchange program, a student must pass two consecutive stages. The probability of passing the first stage is $0.7$. If they pass the first stage, 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 student passes the second stage.

2. Given that the student has passed the second stage, what is the probability that they had NOT passed the first stage originally?

A health center carries out a screening test for a food intolerance that affects $2 \%$ of the population. The test is not perfect: it tests positive in $98 \%$ of 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 efficacy of the test for mass screening in populations where the prevalence of the condition is very low.

In a Year 13 class, $60 \%$ of the students are female. It is known that $80 \%$ of the female students passed the last mathematics exam, while the pass rate among male students was $x \%$.

If the probability that a randomly chosen person from the class passed the exam is $0.74$, calculate the value of $x$ (pass rate among male students). Then, calculate the probability that a randomly chosen person is male, given that they failed the exam.