Exercises: Bayes' theorem

On a streaming platform, $60\,\%$ of the series are original productions, whereas the rest are from external production companies. It is known that $15\,\%$ of the original series are sci-fi, whereas for external series this percentage rises to $25\,\%$. If we choose a series at random and it turns out to be sci-fi, what is the probability that it is an original production of the platform?

A sneaker brand manufactures its models in two different factories: $70\,\%$ in factory $A$ and $30\,\%$ in factory $B$. Quality control has detected that $2\,\%$ of the sneakers from factory $A$ have a stitching defect, whereas in factory $B$ defects appear in $5\,\%$ of production. A customer buys a pair of sneakers and discovers that they have a stitching defect. Calculate the probability that they were manufactured in factory $B$.

In a secondary school, $40\,\%$ of Year 13 students are in the Science stream and $60\,\%$ in Social Sciences. It has been observed that $85\,\%$ of Science students and $90\,\%$ of Social Sciences students pass English. If we choose a student at random who is known to have passed English, what is the probability that they belong to the Science stream?

An influencer splits their posts between two social networks: Instagram ($75\,\%$) and TikTok ($25\,\%$). It is known that $40\,\%$ of their photos on Instagram receive more than 10 000 likes, whereas on TikTok only $20\,\%$ of their videos exceed that figure. If we receive a notification that one of their posts has exceeded 10 000 likes, what is the probability that it is a TikTok video?

An email filter automatically classifies messages as spam. The filter correctly marks $98\,\%$ of emails that are truly spam as spam. However, it also sends $4\,\%$ of legitimate emails to the spam folder. It is estimated that $10\,\%$ of incoming email to an inbox is spam. If a randomly chosen email has been marked as spam, calculate the probability that it is actually spam.

A video game shop sells consoles from two brands: "N-Switch" and "PlayPro". $65\,\%$ of sales are PlayPro and the rest are N-Switch. Technical support reports that $3\,\%$ of PlayPro consoles are returned due to a manufacturing defect, whereas for N-Switch consoles the return percentage is $2\,\%$. If a customer returns a defective console, what is the probability that it is a PlayPro?

To get to the center, a student can go by bus ($80\,\%$ of the time) or by bicycle ($20\,\%$). When traveling by bus, they arrive late $10\,\%$ of the time due to traffic. When traveling by bicycle, they only arrive late $5\,\%$ of the time. If on a given day the student arrives late for class, calculate the probability that they traveled by bus.

In a software development company, $70\,\%$ of the programmers use the Python language and $30\,\%$ use Java. It is estimated that $10\,\%$ of the code written in Python contains critical syntax errors, while in Java this percentage is $15\,\%$. If a critical syntax error is detected in a piece of code, what is the probability that it was written in Python?

At a health center, a study is being conducted on the efficacy of a diagnostic test for a food allergy that affects $8\,\%$ of the youth population. Clinical data indicate that if a person suffers from the allergy, the test gives a positive result in $95\,\%$ of cases. On the other hand, if the person does not suffer from the allergy, the test gives a "false positive" (positive result without being ill) in $4\,\%$ of instances. If a young person is chosen at random and their test result is positive, calculate the probability that they actually suffer from that allergy.

A social network uses an artificial intelligence algorithm to detect hate speech comments. It is known that $5\,\%$ of published comments are actually inappropriate. The algorithm correctly identifies $98\,\%$ of comments that are truly inappropriate as "inappropriate". However, it wrongly labels $3\,\%$ of respectful comments as "inappropriate". If the system has just blocked a comment for marking it as "inappropriate", what is the probability that the comment was actually respectful?

A last-mile logistics company delivers packages in a city using three types of vehicles: electric vans ($60\,\%$), motorbikes ($30\,\%$), and cargo bicycles ($10\,\%$). The probability that a package arrives late is $2\,\%$ if delivered by van, $5\,\%$ if by motorbike, and $8\,\%$ if by bicycle. If a customer calls to complain because their package has arrived late, what is the probability that the courier was on a cargo bicycle?

At a college preparing students for university entrance exams, $40\,\%$ of the students are enrolled in the Science stream, $35\,\%$ in Social Sciences, and $25\,\%$ in Humanities. At the end of the course, $92\,\%$ of Science students, $88\,\%$ of Social Sciences students, and $80\,\%$ of Humanities students pass the entrance exam. Choosing a student at random from the college who is known to have passed the university entrance exam, calculate the probability that they belong to the Social Sciences stream.

A manufacturer of electronic components for smartphones has three production lines: A, B, and C. Line A produces $45\,\%$ of the components, line B produces $30\,\%$, and line C produces $25\,\%$. It is known that the probability of a component being defective is $1\,\%$ on line A, $2\,\%$ on line B, and $3\,\%$ on line C. If a component is selected at random and turns out to be defective, determine with justification which production line is most likely to have manufactured it.

In a city, $70\,\%$ of citizens use a mobile app to pay for public transportation, whereas the rest use a physical card. It is observed that $15\,\%$ of those using the app have reading issues at the ticket validator, compared to $5\,\%$ of those using a physical card. If a validator emits a reading error signal, what is the probability that the user is trying to pay with the mobile app?

A recycling center processes containers coming from two districts of the city, North and South. $60\,\%$ of the containers come from the North district. At the processing plant it is detected that $10\,\%$ of containers from the North are misclassified by citizens, whereas in the South district this percentage rises to $15\,\%$. If the conveyor belt operator selects a container at random and finds that it is correctly classified, what is the probability that it comes from the North district?

An e-sports team plays its tournament matches under two network latency conditions: "Low Latency" ($P=0.75$) and "High Latency" ($P=0.25$). When latency is low, the team wins $80\,\%$ of its matches. However, if latency is high, their probability of victory drops to $40\,\%$. Given that the team has just won a match, what is the probability that they were playing with high latency?

A tech multinational uses three different suppliers for the microchips in its new gaming laptops. Supplier $A$ supplies $45\%$ of the chips, supplier $B$ supplies $35\%$, and the remainder come from supplier $C$. It is known from previous quality checks that $1\%$ of the chips from $A$ are defective, whereas for $B$ and $C$ these percentages are $2\%$ and $3\%$, respectively. A user buys a laptop and, after a month of use, suffers a critical failure due to the microchip. Calculate the probability that the defective chip was supplied by supplier $B$.

A social media algorithm classifies content into three categories: "Viral" ($15\%$), "Trending" ($35\%$), and "Standard" ($50\%$). The probability that a user stops scrolling and watches a video in full is $80\%$ if it is Viral, $40\%$ if it is Trending, and only $10\%$ if it is Standard. If we observe that a user has just finished watching a video in full, what is the probability that the algorithm had not originally classified it as Viral?

A precision parts factory uses a computer vision machine to inspect production quality. It is known that only $0.5\%$ of the manufactured parts have a defect. The machine correctly detects $96\%$ of defective parts, but also erroneously marks $2\%$ of non-defective parts as defective. If the machine has just marked a part as defective, calculate the probability that the part is actually in perfect condition.

A courier company uses electric vans ($60\%$) and combustion vans ($40\%$). It has been observed that $5\%$ of journeys made by electric vans suffer delays due to the need for intermediate charging, whereas delays in combustion vans occur in $8\%$ of cases due to traffic problems. Given that a package has arrived late, determine the probability that it was transported by an electric van.

Three streaming channels ($N$, $H$, and $D$) share the subscriber market in a city in proportions of $40\%$, $30\%$, and $30\%$, respectively. The probability that a subscriber of $N$ cancels their subscription this month is $0.05$; for $H$ it is $0.08$, and for $D$ it is $0.12$. If we choose at random a citizen who has just cancelled their subscription to one of these channels, what is the probability that they were a customer of $D$?

A biometric security system uses facial recognition or fingerprint. $70\%$ of the time facial recognition is used and $30\%$ fingerprint is used. The system fails to identify an authorised user $2\%$ of the time with facial recognition and $1\%$ with fingerprint. On the other hand, it allows access to an intruder $0.5\%$ of the time with facial recognition and $0.1\%$ with fingerprint. Given that the system has denied access to an authorised user, calculate the probability that they were trying to enter using facial recognition.

In a study on teenage reading habits, the sample was divided into three groups according to daily screen time: Low ($20\%$), Medium ($50\%$), and High ($30\%$). The probability that a young person in the Low group reads at least one book a month is $0.7$; for the Medium group it is $0.4$, and for the High group it is $0.15$. If we select a teenager at random and they state that they have not read any books in the last month, calculate the probability that they belong to the "High" screen time group.

A factory has two machines, $M_1$ and $M_2$, that produce the same part. $M_1$ produces three times as many parts as $M_2$. The percentage of defective parts from $M_1$ is $k\%$ and from $M_2$ is $4\%$. It is known that if a part is chosen at random and turns out to be defective, the probability that it comes from machine $M_2$ is $4/13$. Calculate the value of $k$ and then the probability that a randomly chosen part is defective.