4.13 Bayes’ theorem questions for IB Maths AA

In a population, $\frac{1}{50}$ of people have a particular condition, $D$. A screening test for $D$ gives a positive result for $\frac{9}{10}$ of the people who have the condition and for $\frac{1}{10}$ of the people who do not have the condition.

A person is chosen at random from the population and tested.

A café buys all of its coffee beans from three roasters, A, B and C. Of the bags of beans it buys, 45% come from roaster A, 35% from roaster B and 20% from roaster C.

The probability that a bag is stale when it is opened is 0.03 for roaster A, 0.05 for roaster B and 0.08 for roaster C.

A bag of beans is selected at random and opened.

This question asks you to investigate whether a contestant in a game should change their choice after receiving information from a host.

In a game, a prize is placed behind one of three closed doors, numbered 1, 2 and 3. The prize is equally likely to be behind each door. A contestant chooses door 1. The host, who knows where the prize is, then opens one of the other two doors, always choosing a door that does not hide the prize. The contestant may then either stay with door 1 or switch to the other closed door.

If the prize is behind door 1, the host opens door 3 with probability $q$ and door 2 with probability $1-q$, where $0\le q\le 1$.

Let $D_i$ be the event that the prize is behind door $i$, for $i=1,2,3$, and let $H_3$ be the event that the host opens door 3.

Box X contains 3 red counters and 1 blue counter. Box Y contains 2 red counters and 3 blue counters. Box Z contains 1 red counter and 4 blue counters.

Pedro rolls a fair six-sided die. If he rolls a 1 he chooses box X, if he rolls a 2 or a 3 he chooses box Y, and otherwise he chooses box Z. He then takes one counter at random from the chosen box.

A bag contains three dice that look identical. When die A is rolled, the probability of a six is $\frac{1}{6}$. When die B is rolled, the probability of a six is $\frac{1}{3}$. When die C is rolled, the probability of a six is $\frac{1}{2}$.

Mei takes one of the dice from the bag at random and rolls it 10 times. Let $X$ be the number of sixes she obtains. The results of the rolls are independent.

This question asks you to investigate how the probability that a component is faulty changes as the results of repeated tests are obtained.

A factory makes electronic components. Each component is either faulty ($F$) or good ($F^{\mathrm{\prime }}$). A test on a component gives one of two results, "fail" or "pass". The test has accuracy $a$, where $\frac{1}{2}<a<1$: a faulty component fails the test with probability $a$, and a good component passes the test with probability $a$.

When a component is tested more than once, the results are independent of each other, given whether or not the component is faulty.

For any event $E$ with $0<\mathrm{P}(E)<1$, the odds of $E$ are defined as $\mathrm{O}(E)={\displaystyle \frac{\mathrm{P}(E)}{\mathrm{P}(E^{\mathrm{\prime }})}}$. Similarly, $\mathrm{O}(E\mid R)={\displaystyle \frac{\mathrm{P}(E\mid R)}{\mathrm{P}(E^{\mathrm{\prime }}\mid R)}}$ for an event $R$.