AHL 4.13—Bayes theorem Questionbank

A university offers three language courses: French (45% of students), Spanish (35%), and German (20%). The probability of passing is $80 \%$ for French, $75 \%$ for Spanish, and $70 \%$ for German. A student is known to have passed their course.

A school has three bus routes: Route 1 ( $40 \%$ of students), Route 2 ( $35 \%$ ), and Route 3 ( $25 \%$ ). The probabilities of arriving late are $12 \%$ for Route 1, $10 \%$ for Route 2, and $15 \%$ for Route 3. A student arrives late.

A hospital uses three diagnostic tests ( $\mathrm{A}, \mathrm{B}$, and C ) to detect a rare disease. Test A is used $40 \%$ of the time, Test B $35 \%$ of the time, and Test C $25 \%$ of the time. The probabilities of a false positive result are $5 \%$ for Test A, $8 \%$ for Test B, and $10 \%$ for Test C. The disease is present in $2 \%$ of the population. If a patient receives a positive test result, calculate the probability that they were tested with Test C.

A laboratory tests for a genetic condition using two methods: Method X ( $70 \%$ of tests) and Method Y ( $30 \%$ of tests). Method X has a true positive rate of $92 \%$ and a false positive rate of $7 \%$. Method Y has a true positive rate of $88 \%$ and a false positive rate of $10 \%$. The condition is present in $4 \%$ of the tested population. If a test is positive, find the probability that Method X was used.

A telecommunications company uses three types of internet routers: Type A (45% of installations), Type B (30%), and Type C (25%). The failure rates due to connectivity issues are $6 \%$ for Type A, $9 \%$ for Type B, and $12 \%$ for Type C. If a router fails due to connectivity issues, calculate the probability that it is Type B.

A clinic tests for a condition using three methods: Method X (50% of tests), Method Y (30%), and Method Z (20%). The true positive rates are $98 \%$ for Method X, $95 \%$ for Method Y , and $90 \%$ for Method Z. The false positive rates are $6 \%$ for Method X, $8 \%$ for Method Y, and $10 \%$ for Method Z. The condition affects $3 \%$ of the population.

A factory produces components using three machines: Machine P (50% of production), Machine Q (35%), and Machine R (15%). The defect rates are $5 \%$ for Machine P, $8 \%$ for Machine Q, and $10 \%$ for Machine R. If a component is defective, it undergoes a repair process with a $20 \%$ success rate, independent of the machine.

A factory produces three types of batteries: Type X (50% of production), Type Y (30%), and Type Z (20%). The defect rates are $4 \%$ for Type X, $6 \%$ for Type Y, and $10 \%$ for Type Z. A battery is selected at random and found to be defective. Find the probability that it is Type Y.

A quality control system inspects products from three machines: Machine P (50% of products), Machine Q (30%), and Machine R (20%). The defect rates are 3% for Machine P, $5 \%$ for Machine Q, and $7 \%$ for Machine R. A defective product is found, and it is known that $10 \%$ of defective products are repaired successfully.

A museum uses three types of tickets: Adult (55% of sales), Student (30%), and Senior ( $15 \%$ ). The probabilities of a ticket being used for a special exhibit are $20 \%$ for Adult, $25 \%$ for Student, and $30 \%$ for Senior. If a ticket is used for the special exhibit, calculate the probability that it is a Student ticket.