Bayes' Theorem

To get to the centre, a student can go by bus ($80\,\%$ of the time) or by bicycle ($20\,\%$). When travelling by bus, they arrive late $10\,\%$ of the time due to traffic. When travelling 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 travelled by bus.

A trainer 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 trainers from factory $A$ have a stitching defect, whereas in factory $B$ defects appear in $5\,\%$ of production. A customer buys a pair of trainers and discovers that they have a stitching defect. Calculate the probability that they were manufactured in factory $B$.

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.

At a health centre, 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.

In a city, $70\,\%$ of citizens use a mobile app to pay for public transport, 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 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.

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?

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 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 $B$.