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 behaviour 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 secondary school, a survey on extracurricular activities was conducted. The data show that $45\,\%$ of the students regularly practice sports ($D$), $30\,\%$ attend art workshops ($A$), and $15\,\%$ participate in both activities. Lucía, a 2nd-year Bachillerato student, wants to calculate the probability that a randomly chosen student participates in art workshops given that they practice sports ($P(A|D)$), and writes the following in her notebook:

"Since $15\,\%$ of students do both things out of the total, we have $P(A|D) = 0,15$. Furthermore, the probability of practicing sports given that they attend art is exactly the same, $P(D|A) = 0,15$, because the intersection is commutative ($A \cap D = D \cap A$)".

1. a) Pinpoint the two conceptual errors made by Lucía and explain why her reasoning contradicts the idea of "reduction of the sample space" inherent to conditional probability.
2. b) Correct the approach and calculate, with full justification, the correct values of $P(A|D)$ and $P(D|A)$.

In a language academy that prepares students for the official B2 level English certification, $70\,\%$ of enrolled students regularly attend in-person classes ($C$), while the remaining $30\,\%$ prepare through self-study on their own ($\overline{C}$). According to the academy's historical exam sessions, $80\,\%$ of those who attend in-person classes pass the official exam, but only $40\,\%$ of those who study on their own do.

1. a) Before carrying out any analytical calculation, make a reasoned estimate: if we know that an arbitrary student has passed the exam ($A$), should the probability that they belong to the in-person attendance group, $P(C|A)$, be greater than, less than, or equal to the initial $70\,\%$? Justify your estimate intuitively.
2. b) Model the situation using compound probability calculations to formally obtain the exact value of $P(C|A)$. Then, compare the result with your initial estimate and argue why the posterior information modifies the certainty we had about the initial event.

In a survey carried out on 200 students at a secondary 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 centre 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?

Based on an adaptation of data from the Labour Force Survey (EPA) of the National Statistics Institute (INE) regarding the young Spanish population aged 18 to 29, the following probabilities concerning their employment and housing situation are obtained:

• The probability that a young person is emancipated (lives outside the family home) is $P(E) = 0,16$.

• The probability that they have a permanent employment contract is $P(I) = 0,45$.

• The probability that they are not emancipated and at the same time do not have a permanent contract amounts to $51\,\%$ ($P(\overline{E} \cap \overline{I}) = 0,51$).

1. a) Use De Morgan's laws to calculate the probability that a young person is emancipated, has a permanent contract, or both ($P(E \cup I)$).
2. b) Deduce the probability that a randomly selected young person simultaneously meets both conditions: being emancipated and having a permanent contract ($P(E \cap I)$).
3. c) If a young person who has a permanent contract is chosen at random, calculate the conditional probability that they are emancipated, $P(E|I)$. Interpret the result obtained by comparing it with the marginal probability $P(E)$ and conclude whether having a permanent contract promotes youth emancipation or not.

An online sales platform distributes its orders through two delivery methods: «Standard Shipping» ($S$) and «Express Shipping» ($U$). $60\,\%$ of purchases are handled via standard shipping and $40\,\%$ via express shipping. An order may experience delivery delay issues ($R$). It is known from the logistics operator's records that, among express shipments, the probability of experiencing a delay is only $P(R|U) = 0,02$.

The quality control department has set as a priority objective that, among all shipments that arrive late to customers, the probability that they correspond to the express method does not exceed $10\,\%$ (that is, it requires that $P(U|R) = 0,10$).

Determine what the maximum allowable delay rate for standard shipments must be, that is, the value of the conditional probability $P(R|S)$, for the company to meet exactly the established quality standard.

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 colour, 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 programme, 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 centre 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.

In a public health research study on a seasonal respiratory infection in a population of 1000 people, the following clinical information is available:

• 200 people present symptoms compatible with the infection ($S$) and the rest are asymptomatic ($\overline{S}$).

• $75\,\%$ of people with symptoms test positive in a rapid diagnostic test ($T$).

• Among asymptomatic people, $10\,\%$ test positive (false positive).

The medical team needs to determine the probability that a randomly chosen person who tested positive is actually asymptomatic, that is, $P(\overline{S}|T)$.

1. a) Propose the resolution of this situation through two different methodological strategies:

• Method 1: Construction of a probabilistic tree diagram and rigorous application of the axiomatic formulas for conditional probability and intersection.

• Method 2: Preparation of a contingency table of absolute frequencies distributing the 1000 people into the corresponding cells.

1. b) Execute both methods until obtaining the value of $P(\overline{S}|T)$ and verify that they lead to the same numerical result.
2. c) Provide a critical evaluation comparing both approaches: which of the two methods do you consider more pedagogical and less prone to comprehension errors for communicating this result to a team of public managers unfamiliar with formal statistics? Justify your answer.

In a coastal region, the State Meteorological Agency (AEMET) activates the «Torrential rain storm alert» protocol ($A$) on $8\,\%$ of autumn days. Historical records from the local observatory indicate that it actually rains torrentially ($L$) on $5\,\%$ of autumn days. In addition, the data reveal that the probability of torrential rain on a day when the prior alert has been activated is $P(L|A) = 0,50$.

1. a) Prove using the formal definition of independent events whether the activation of the alert ($A$) and the actual occurrence of torrential rain ($L$) are independent events. Explain the operational meaning that this conclusion has for Civil Protection services.
2. b) Calculate the probability that the weather alert was activated knowing that on a given day torrential rains were ultimately not recorded, that is, $P(A|\overline{L})$.
3. c) During a press conference, a municipal spokesperson makes the following criticism: «Given that $P(L|A) = 0,50$, the AEMET warning system fails half the time; therefore, the alert completely lacks informative utility». Refute this claim mathematically by comparing the conditional probability with the prior probability of torrential rain without an alert, and reflect on the asymmetric cost of the two possible types of errors (false alarm versus unexpected storm).