IB Mathematics Analysis and Approaches Topic AHL…

A factory has two machines, $A$ and $B$ , producing electrical components. Machine $A$ produces $60\%$ of all components and machine $B$ produces $40\%$ . Among the components produced by machine , are defective; among those produced by machine , are defective.

(a) Find the probability that a randomly selected component is defective.

(b) Given that a randomly selected component is defective, find the probability that it was produced by machine $A$ .

(e) The hospital sees $1000$ patients in a month, assumed independent.

Find the expected number of patients misdiagnosed in a month.

(f) The hospital sees $1000$ patients in a month, assumed independent.

Find the standard deviation of the number of patients misdiagnosed in a month.

(g) Find the smallest integer $n$ such that the probability of obtaining at least one misdiagnosis among $n$ independently chosen patients exceeds $0.999$ .

(e) Find the expected number of short-lived bulbs in the sample.

A factory has three machines $A$ , $B$ and $C$ , producing $40\%$ , $35\%$ and $25\%$ of the total output respectively. The probability that an item is defective is $0.02$ , $0.03$ and $0.06$ for machines $A$ , $B$ and $C$ respectively.

(a) Find the probability that a randomly selected item is defective.

(b) Given that an item is defective, find the probability that it was produced by machine $C$ .

(d) Two items are selected at random, independently.

Find the probability that exactly one of the two items is defective.

In a hospital, three doctors $A$ , $B$ and $C$ see patients in proportions $0.5$ , $0.3$ and $0.2$ respectively. The probability that a patient is correctly diagnosed by doctors $A$ , $B$ and $C$ is $0.92$ , $0.85$ and $0.78$ respectively.

(a) Find the probability that a randomly selected patient is correctly diagnosed.

(b) Given that a patient is misdiagnosed, find the probability that the patient was seen by doctor $C$ .

Find the probability that exactly two of these patients are correctly diagnosed.

(d) Ten patients are independently chosen at random.

Find the probability that at least four of these patients are misdiagnosed.

A company manufactures lightbulbs at two factories, $F_1$ and $F_2$ . Factory $F_1$ produces $65\%$ of all bulbs and factory $F_2$ produces the remaining $35\%$ . The lifetime, in hours, of a bulb produced at $F_1$ is normally distributed with mean $920$ and standard deviation $110$ . The lifetime, in hours, of a bulb produced at $F_2$ is normally distributed with mean $880$ and standard deviation $80$ .

A bulb is considered short-lived if its lifetime is less than $750$ hours.

Twenty bulbs are selected at random.

(a) Find the probability that a bulb produced at $F_1$ is short-lived.

(b) Find the probability that a randomly selected bulb is short-lived.

(d) Find the probability that exactly $3$ of them are short-lived.

A university has three language classes available: Italian ( $0.50$ of students), Japanese ( $0.30$ ), and Arabic ( $0.20$ ). The probability of passing is $0.76$ for Italian, $0.80$ for Japanese, and $0.90$ for Arabic.

(a) a probability tree diagram showing the language chosen and the resulting pass/fail outcomes.

(b) the probability that a randomly chosen student passes their course.

(d) If a student passes, find the probability that they were not enrolled in Italian.

IB Mathematics Analysis and Approaches AHL 4.13—bayes Theorem Questionbank

IB Mathematics Analysis and Approaches AHL 4.13—bayes Theorem Questionbank

IB Mathematics Analysis and Approaches AHL 4.13—bayes Theorem Questionbank

IB Mathematics Analysis and Approaches AHL 4.13—bayes Theorem Questionbank

IB Mathematics Analysis and Approaches (AA) Topic AHL 4.13—bayes Theorem Questionbank