Exercises: Total probability and Bayes' theorem

In a school, Year 13 students can choose between two platforms to carry out their work in the cloud: $65\%$ use Google Workspace and the rest use Microsoft 365. It is known that $20\%$ of Google users use Artificial Intelligence tools to write drafts, whereas in Microsoft $15\%$ do so.

Calculate the probability that a student chosen at random uses Artificial Intelligence for their drafts.

A supermarket chain sells yoghurts from two different suppliers. $40\%$ comes from Supplier A and $60\%$ from Supplier B. Quality control indicates that $5\%$ of yoghurts from Supplier A have packaging defects, whereas for Supplier B this defect only appears in $2\%$ of cases.

If we buy a yoghurt and find that it has a packaging defect, what is the probability that it came from Supplier A?

A TikTok content creator observes that their short videos are viewed in two ways: via the recommendation algorithm ($80\%$ of the time) or via direct search ($20\%$). The probability that a user clicks "Like" is $0.10$ if the video arrives by recommendation, and $0.30$ if the user searched for it directly.

Find the probability that any user viewing the video clicks "Like".

In an esports tournament, a player reaches the final. They can choose between two strategies: Offensive (chosen $70\%$ of the time) or Defensive ($30\%$). If they play offensively, they win the game with a probability of $0.6$, whereas if they play defensively, their probability of victory is $0.4$.

Given that the player won the game, calculate the probability that they used a defensive strategy.

A courier service uses three routes to deliver parcels in a city: North Route ($50\%$), South Route ($30\%$) and Central Route ($20\%$). Delivery delays occur $2\%$ of the time on the North route, $4\%$ on the South route and $10\%$ on the Central route.

Determine the total percentage of packages delivered late in this city.

A study is carried out on a new fitness app. It is estimated that $1\%$ of teenagers in a city use it. The app has a "daily challenge" feature that is successfully completed $95\%$ of the time if the user is consistent, but if they are not ($99\%$ of the population does not use the app), the probability of completing a similar challenge by chance is only $0.05$.

If we select a young person at random and see that they have successfully completed the challenge, what is the probability that they are a consistent user of the app?

In a design company, $60\%$ of the employees are Senior Designers and $40\%$ are Juniors. In the last advertising campaign, $90\%$ of the Seniors used vector design software, whereas among the Juniors only $70\%$ did so.

If we choose an employee at random, calculate the probability that they did not use vector design software.

An electric car dealership sells three models: Compact ($40\,\%$), Sedan ($35\,\%$) and SUV ($25\,\%$). It has been observed that $3\,\%$ of Compacts, $2\,\%$ of Sedans and $5\,\%$ of SUVs need a software update in the first month. The figure shows the tree diagram of the situation.

If a customer calls because their car has needed a software update in the first month, which model is most likely to have been purchased? Justify your answer by calculating the corresponding probabilities.

Seán is solving a quality control problem for an olive-growing cooperative in Cork. The olive supply comes from two estates: estate $F_1$ contributes $60\,\%$ of the harvest and estate $F_2$ contributes $40\,\%$. The percentage of olives not suitable for producing extra virgin olive oil is $5\,\%$ in estate $F_1$ and $10\,\%$ in estate $F_2$. The problem asks to calculate the probability that an unsuitable olive comes from estate $F_1$. Seán writes on his exam:

"$P(F_1 | \text{Unsuitable}) = P(F_1) \times P(\text{Unsuitable} | F_1) = 0.60 \times 0.05 = 0.03$. Therefore, there is a $3\,\%$ probability that it comes from estate $F_1$. It makes sense because it is a number less than $1$".

1. a) Identify and explain in detail the two conceptual errors Seán has made (one related to the confusion between intersection and conditional probability, and another with Bayes' Theorem).
2. b) Correct the exercise by rigorously calculating the requested probability.

According to passenger mobility reports on an Irish rail corridor, $50\,\%$ of users travel on High-Speed services (Irish Rail InterCity), $30\,\%$ on Mid-Distance (Regional), and the remaining $20\,\%$ on Commuter Rail (DART/Commuter). Punctuality records indicate that the probability of suffering a delay greater than $15$ minutes is $2\,\%$ on High-Speed, $6\,\%$ on Mid-Distance, and $10\,\%$ on Commuter Rail.

1. a) Justify that the types of train form a complete system of events and clearly define the events involved.
2. b) Calculate the probability that a passenger chosen at random on that corridor arrives at their destination with a delay of more than $15$ minutes.
3. c) If a randomly selected passenger suffered a delay greater than $15$ minutes, calculate the probability that their trip was on High-Speed. Comment on the result: why is this probability relatively low even though half of the tickets sold correspond to that service?

In an esports high-performance centre, players train on three different platforms: PC, console and mobile devices. $50\ \%$ of players use PC, $30\ \%$ use console and the rest mobile devices. It has been observed that performance decreases due to eye strain in $8\ \%$ of PC players, $5\ \%$ of console players and $12\ \%$ of mobile users.

1. a) Calculate the probability that a player chosen at random suffers from eye strain.

1. b) If we select a player and see that they suffer from eye strain, what is the probability that they usually train with a mobile device?

A cybersecurity company analyses the email traffic of a tech startup. $70\ \%$ of emails received are internal, $20\ \%$ come from known clients and the remaining $10\ \%$ are external emails from an unknown source. The security filter flags as "suspicious" $1\ \%$ of internal emails, $4\ \%$ of client emails and $25\ \%$ of unknown emails.

1. a) Determine the probability that an email chosen at random is flagged as suspicious by the system.

1. b) If an employee receives a suspicious email notification, what is the probability that the email is actually from an unknown source?

In an electric car battery assembly plant, two production lines, $L_1$ and $L_2$, are used. Line $L_1$ produces $60\ \%$ of the batteries and $L_2$ the rest. It is known that $2\ \%$ of the batteries from $L_1$ have sealing defects, whereas in $L_2$ this defect appears in $5\ \%$ of the units.

1. a) If a battery is chosen at random from the total production, what is the probability that it is not defective?

1. b) If a quality control check detects a defective battery, which line is more likely to have manufactured it? Justify your answer by calculating both posterior probabilities.

A study on sustainable mobility in a city indicates that $45\ \%$ of journeys are made by public transport, $35\ \%$ by bicycle or e-scooter, and $20\ \%$ by private car. The probability of arriving late for work is estimated to be 0.15 if using public transport, 0.05 if using bicycle/e-scooter, and 0.30 if using private car (due to traffic jams).

1. a) Calculate the probability that a citizen chosen at random arrives on time for work.

1. b) Given that a worker arrived late, what is the probability that they travelled in their private car?

A pharmaceutical laboratory develops a rapid test to detect a vitamin deficiency affecting $5\ \%$ of the young population. Clinical trials show that the test gives a positive result in $96\ \%$ of people who actually have the deficiency. However, it also gives a "false positive" in $2\ \%$ of healthy people.

1. a) What percentage of the total population will test positive?

1. b) If a young person takes the test and the result is positive, what is the actual probability that they suffer from this vitamin deficiency? Comment on whether you find the result high or low in relation to the reliability of the test.

In a Year 13 subject, students can choose to do a final project on three topics: Robotics, Programming or Renewable Energy. $25\ \%$ choose Robotics, $40\ \%$ Programming and $35\ \%$ Renewable Energy. The probability of obtaining a Distinction grade is 0.20 in Robotics, 0.15 in Programming and 0.10 in Renewable Energy.

1. a) Find the probability that a student in the class, chosen at random, does not obtain a Distinction.

1. b) Given that a student obtained a Distinction, what is the probability that their project was on Robotics?

A music streaming service classifies its users into three categories: "Free" (with ads), "Premium" and "Family". $60\ \%$ are Free users, $30\ \%$ Premium and $10\ \%$ Family. It is known that $80\ \%$ of Free users listen for more than 2 hours daily, whereas this percentage is $90\ \%$ for Premium users and $95\ \%$ for Family users.

1. a) If a user is chosen at random, what is the probability that they listen to music for 2 hours or less per day?

1. b) If we select a user who listens to more than 2 hours of music daily, what is the probability that they belong to the "Premium" category?

In a university entrance exam, a student encounters a multiple-choice question with 4 possible answers, of which only one is correct. The probability of the student knowing the correct answer is 0.6. If they know it, they answer correctly for sure. If they do not know it, they choose one of the 4 options at random.

1. a) What is the probability that the student answers the question correctly?

1. b) If the examiner sees that the marked answer is correct, what is the probability that the student actually knew the subject matter and did not answer at random?

In a preventative traffic campaign conducted by the RSA on national roads, it is estimated that $1\,\%$ of drivers circulate having consumed illicit substances ($S$). The preliminary saliva test used at checkpoints has a sensitivity of $98\,\%$ (if a driver has consumed, it tests positive with probability $0.98$), but it has a false positive rate of $4\,\%$ (in drivers who have not consumed, the test mistakenly tests positive with probability $0.04$).

1. a) Before carrying out any numerical calculation: if a driver tests positive at the checkpoint, intuitively estimate and briefly reason whether the probability that they have actually consumed substances will be closer to $90\,\%$, to $50\,\%$, or below $25\,\%$.
2. b) Apply Bayes' Theorem to rigorously calculate $P(S | \text{Positive})$.
3. c) Compare the obtained value with your estimate from part a). Explain what probabilistic phenomenon causes this result and why the regulations require sending a second sample to a toxicological laboratory before processing the sanction.

A parcel logistics centre located in the Corredor del Henares distributes its shipments using two automated sorting lines, $L_1$ and $L_2$. Line $L_1$ handles a proportion $p$ of the total parcel volume ($0 < p < 1$), while line $L_2$ sorts the remaining fraction ($1-p$). Due to mechanical wear, line $L_1$ damages $1\,\%$ of the parcels it processes, whereas line $L_2$, with older technology, damages $4\,\%$.

1. a) Express in terms of $p$ the probability that a parcel randomly selected at the end of the process turns out to be damaged, $P(D)$.
2. b) Technical management establishes that, in order not to overload line $L_1$'s inspection department, of all parcels detected as damaged, at most $20\,\%$ must come from said line ($P(L_1 | D) \le 0.20$). Set up the corresponding inequality and determine what minimum percentage of the total parcel traffic must be assigned to line $L_2$ to meet this quality criterion.

A cybersecurity company uses three artificial intelligence systems ($S_1$, $S_2$ and $S_3$) to detect unauthorized access attempts on a corporate network. System $S_1$ handles $45 \%$ of the alerts, $S_2$ handles $35 \%$ and $S_3$ handles the rest.

It is known from previous audits that, among the alerts issued by each system, the following are false alarms (a legitimate access is reported as an attack): $2 \%$ of those from $S_1$, $5 \%$ of those from $S_2$ and $8 \%$ of those from $S_3$. The remaining alerts from each system correspond to real attacks. Moreover, the probability that a system ignores a real attack (false negative) is $1 \%$ in all cases; this is information you will not need for this problem.

At a given moment, an alert goes off on the control panel indicating an attack.

1. a) Represent the situation with a tree diagram, detailing the probabilities of each branch.

1. b) Calculate the probability that the alert issued is actually a false alarm.

1. c) If it is confirmed that the alert was a false alarm, what is the probability that it was generated by system $S_3$?

An automated assembly line produces racing drones. The factory uses two types of engine: the Alpha engine, which is installed in $60 \%$ of the drones, and the Beta engine, which is installed in the rest. Due to the high precision required, some engines have calibration faults.

It is known that $5 \%$ of the Alpha engines are defective. After conducting a quality test at the end of the line, the overall probability of finding a drone with a defective engine is observed to be $4.4 \%$.

1. a) Calculate the probability that an engine is defective given that it is of type Beta.
2. b) A drone is chosen at random and the quality test indicates that the engine works correctly. What is the probability that the drone has an Alpha engine fitted?

In a high-performance sports centre, athletes undergo doping control. $1 \%$ of the athletes consume a banned substance. The test used has a reliability of $98 \%$ (probability of testing positive if the substance has been consumed). However, the test also produces $3 \%$ "false positives" (testing positive in athletes who have not consumed anything).

1. a) If an athlete tests positive in the control, what is the probability that they have actually consumed the substance?
2. b) The committee decides to perform a second independent test on the same athlete if the first one comes back positive. If this second test (with the same reliability) also turns out positive, what is the probability now that the athlete is guilty? Justify whether the confidence in the verdict has increased significantly.

A study on streaming content consumption reveals that $70 \%$ of young people in a city are subscribed to Platform A, $40 \%$ to Platform B and $20 \%$ to both. A young person is chosen at random.

1. a) Given that they are subscribed to Platform B, what is the probability that they are also subscribed to Platform A? Are the events "Being subscribed to A" and "Being subscribed to B" independent?

1. b) It is known that $15 \%$ of the users of only Platform A prefer documentaries, $25 \%$ of the users of only Platform B also prefer them, and among those who are on both platforms, that preference rises to $50 \%$. Among the young people who are not subscribed to either platform, $10 \%$ prefer documentaries. If a randomly chosen young person prefers documentaries, what is the probability that they are subscribed to both platforms?

In a biotechnology laboratory, three types of bacteria are handled: $T_1$, $T_2$ and $T_3$. There is a container with 100 samples: 20 of $T_1$, 30 of $T_2$ and 50 of $T_3$. The probability of a sample developing antibiotic resistance within 24 hours is $0.2$ for $T_1$, $0.4$ for $T_2$ and an unknown value $p$ for $T_3$.

A sample is drawn at random and observed to have developed resistance after 24 hours. Given that the probability of this sample being of type $T_2$ is $0.375$:

1. a) Determine the value of the probability $p$ associated with type $T_3$ bacteria.
2. b) Using the value of $p$ obtained, calculate the probability that, when choosing a sample at random, it does not develop resistance.

A news channel uses three different sources to obtain information: Agencies ($A$), Correspondents ($C$) and Social Media ($R$). $50 \%$ of the news stories come from $A$, $30 \%$ from $C$ and the rest from $R$.

The probability of a news story being false (fake news) is $0.01$ if it comes from $A$ and $0.05$ if it comes from $C$. It has been detected that, if a news story chosen at random turns out to be false, the probability that it came from Social Media is $0.75$.

1. a) Calculate the probability that a news story coming from Social Media is false.
2. b) Calculate the total probability that a news story chosen at random from this channel is true.

Three machines, $M_1$, $M_2$ and $M_3$, manufacture precision parts. $M_1$ produces twice as many parts as $M_2$, and $M_3$ produces the same number as $M_1$ and $M_2$ combined. The percentage of defective parts is $2 \%$ for $M_1$ and $4 \%$ for $M_2$.

It is known that, when taking a part at random from the total production, the probability of it being defective is $2.5 \%$.

1. a) Calculate the percentage of defective parts produced by machine $M_3$.
2. b) If we take a part and find that it is defective, which machine is most likely to have manufactured it?

In a competitive exam, the syllabus consists of 80 topics. The panel draws two topics at random (without replacement) and the candidate must choose one of them to write on. A candidate has prepared only 45 topics.

1. a) What is the probability that the candidate can choose at least one topic that they know?
2. b) Suppose the candidate passes if the chosen topic is one of those they have prepared. Given that the candidate failed because they could not choose any of the drawn topics, what is the probability that, if the exam had consisted of drawing three topics instead of two, they would have had the opportunity to choose at least one? Justify mathematically whether the probability of passing would increase significantly.

In a public health programme of the Health Service Executive (HSE) in Ireland for the early detection of colorectal cancer in people between $50$ and $69$ years old, the prevalence of the condition in that demographic group is $0.5\,\%$. The faecal occult blood test used has a sensitivity of $80\,\%$ (it tests positive in $80\,\%$ of diseased people) and a false positive rate of $3\,\%$ (it tests positive in $3\,\%$ of healthy people). A patient tests positive in the screening, and we wish to calculate the probability that they actually have the disease, $P(E | +)$.

1. a) Method 1 (Algebra of events): Formally state the partition of the sample space, the prior probabilities, the likelihoods, and solve $P(E | +)$ by applying the Law of Total Probability and Bayes' Theorem.
2. b) Method 2 (Natural frequencies): Model the situation using a hypothetical cohort of $100,000$ people. Construct a $2 \times 2$ contingency table with the integer numbers of healthy/diseased and positive/negative individuals, and deduce directly from it the value of $P(E | +)$.
3. c) Critical evaluation: Compare both methods and justify with sound arguments why various research studies in cognitive psychology and mathematics education argue that Method 2 is much more transparent and reduces diagnostic cognitive bias compared to the abstract algebraic formulation of Method 1.

The IT audit department of an Irish banking institution monitors international transfers to prevent fraud. Transactions originate through three channels: SEPA/EU Channel ($70\,\%$ of total volume), Ordinary International Channel ($20\,\%$), and High-Risk International Channel ($10\,\%$). From historical records, it is known that the probability of a transfer being fraudulent ($F$) is $0.1\,\%$ in the SEPA channel, $0.5\,\%$ in the Ordinary International channel, and an unknown value $k$ ($0 < k \le 1$) in the High-Risk channel.

1. a) If the overall probability that a transfer chosen at random in the bank is fraudulent is exactly $0.0037$ ($0.37\,\%$), calculate the value of parameter $k$ using the law of total probability.
2. b) If the security system intercepts a transfer and it is found to be fraudulent, determine using Bayes' Theorem which channel it is most likely to come from.
3. c) Reason through the following theoretical question: Suppose that the intrinsic fraud rates of each channel remain unchanged (the conditional percentages $P(F|\text{channel})$ from the statement). Would it be possible to modify only the distribution of transaction volume among the three channels so that, for a transfer known to be fraudulent, it is more likely to come from the SEPA Channel than from the other two combined ($P(\text{SEPA}|F) > 0.50$)? Justify your answer by deducing what algebraic condition the new channel market shares would have to satisfy.