Bayes' Theorem

To get to school, a student can go by bus ($80\,\%$ of the time) or by bicycle ($20\,\%$). When they go by bus, they arrive late $10\,\%$ of the days due to traffic. When they go by bicycle, they only arrive late $5\,\%$ of the time. If one day the student arrives late to class, calculate the probability that they went by bus.

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

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

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

In a high school, $40\,\%$ of the 12th grade students are in the Science track and $60\,\%$ in Social Sciences. It has been observed that $85\,\%$ of the Science students and $90\,\%$ of the Social Sciences students pass English class. If we choose a student at random who is known to have passed English, what is the probability that they belong to the Science track?

An e-sports team plays its tournament matches under two network conditions: "Low Latency" ($P=0,75$) and "High Latency" ($P=0,25$). When the latency is low, the team wins $80\,\%$ of its matches. However, if the latency is high, its probability of winning drops to $40\,\%$. Knowing that the team has just won a match, what is the probability that they were playing with high latency?

A test to detect a food allergy gives a positive result in $98\,\%$ of the people who actually suffer from the allergy. However, it also gives a "false positive" (positive without having the allergy) in $4\,\%$ of healthy people. We know that in a population $10\,\%$ of the people are allergic. If a person chosen at random tests positive, calculate the probability that they are actually allergic.

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

At a high-performance sports center, a detection test for a banned substance is carried out. It is known that only $0,5\%$ of the athletes consume this substance. The test has a reliability such that it tests positive in $96\%$ of the cases where the athlete has consumed the substance (sensitivity). However, the test also yields $2\%$ of "false positives" in clean athletes. If an athlete tests positive, calculate the probability that they are actually innocent and have not consumed the substance.

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

A manufacturer of electronic components for smartphones has three production lines: A, B and C. Line A produces $45\,\%$ of the components, B $30\,\%$ and C $25\,\%$. 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 reasoning which production line is most likely to have manufactured it.

A multinational technology company uses three different suppliers for the microchips in its new gaming laptops. Supplier $A$ provides $45\%$ of the chips, supplier $B$ provides $35\%$, and the rest come from supplier $C$. It is known from previous quality controls 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 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 the containers from the North are incorrectly sorted by citizens, whereas in the South district this percentage rises to $15\,\%$. If the conveyor belt operator selects a container at random and checks that it is correctly sorted, what is the probability that it comes from the North district?

In a study on reading habits in teenagers, the sample was divided into three groups according to daily screen time: Low ($20\%$), Medium ($50\%$) and High ($30\%$). The probability of a young person in the Low group reading 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 claim not to have read any books in the last month, calculate the probability that they belong to the "High" screen consumption group.

A factory has two machines, $M_1$ and $M_2$, which 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, subsequently, the probability that a randomly chosen part is defective.