In a group of $200$ students, $118$ study chemistry, $92$ study biology, and $46$ study neither chemistry nor biology. Let $C$ be the event that a student studies chemistry and let $B$ be the event that a student studies biology.

A bag contains $5$ red balls, $3$ blue balls and $2$ green balls. Two balls are selected at random without replacement.

A company sells two types of calculators. A randomly selected calculator is graphing with probability $0.60$. The probability that a graphing calculator is faulty is $0.08$, and the probability that a non-graphing calculator is faulty is $0.03$. Let $G$ be the event that the calculator is graphing and let $F$ be the event that it is faulty.

A box is chosen at random from boxes $A$, $B$ and $C$ with probabilities $\frac{1}{2}$, $\frac{1}{3}$ and $\frac{1}{6}$ respectively. A ball is then selected from the chosen box. The probabilities of selecting a red ball from boxes $A$, $B$ and $C$ are $\frac{1}{4}$, $\frac{1}{2}$ and $\frac{3}{4}$ respectively. Let $R$ be the event that the selected ball is red.

A container is chosen at random. Container $A$ is chosen with probability $\frac{2}{5}$ and container $B$ is chosen with probability $\frac{3}{5}$. A counter is then selected from the chosen container. The probability that a red counter is selected from container $A$ is $\frac{1}{4}$, and the probability that a red counter is selected from container $B$ is $p$. It is given that, if the selected counter is red, the probability that it came from container $B$ is $\frac{3}{4}$.

Events $B$ and $B'$ form a partition of the sample space, with $P(B)=\frac{2}{3}$. An event $A$ is such that $P(A\mid B)=k$ and $P(A\mid B')=\frac{1}{4}$. It is given that $P(B\mid A)=\frac{2}{3}$.

Three factories, $F_1$, $F_2$ and $F_3$, produce components for a company. The probabilities that a randomly selected component was produced by $F_1$, $F_2$ and $F_3$ are $\frac{1}{2}$, $\frac{1}{3}$ and $\frac{1}{6}$ respectively. The probabilities that a component is defective, given that it was produced by $F_1$, $F_2$ and $F_3$, are $\frac{1}{20}$, $\frac{1}{10}$ and $\frac{1}{5}$ respectively. Let $D$ be the event that a randomly selected component is defective.

Two events $A$ and $B$ are independent. It is given that $\mathrm{P}(A)=0.42$ and $\mathrm{P}(A\cup B)=0.70$. Let $\mathrm{P}(B)=p$.

In a computer simulation of $5000$ independent trials, an event $E$ occurred $1738$ times. A theoretical model predicts that the probability of $E$ in one trial is $0.36$.

A disease affects $1.8\%$ of a population. A screening test gives a positive result for $94\%$ of people who have the disease and for $7\%$ of people who do not have the disease. A person is selected at random and tested.

A factory has three machines, $A$, $B$ and $C$. Machine $A$ produces $45\%$ of the items, machine $B$ produces $35\%$ and machine $C$ produces the rest. The probabilities that an item is defective given that it was produced by $A$, $B$ and $C$ are $0.012$, $0.018$ and $0.030$ respectively.

Orders placed with an online shop come from three regions: north, east and south. The probabilities that an order comes from these regions are $0.50$, $0.30$ and $0.20$ respectively. The probabilities that an order is late given that it is from north, east and south are $0.04$, $0.07$ and $0.10$ respectively.

Passengers at an airport join one of three security queues, $Q_1$, $Q_2$ and $Q_3$. The probabilities that a passenger joins $Q_1$, $Q_2$ and $Q_3$ are $0.25$, $0.45$ and $0.30$ respectively. The probabilities that a passenger waits more than $15$ minutes given that they joined $Q_1$, $Q_2$ and $Q_3$ are $0.18$, $0.12$ and $0.08$ respectively.

A bag contains eight tiles, each labelled with one letter from the word ANALYSIS. Three tiles are selected at random, one after another, without replacement.

In a school club of $100$ students, each student may attend drama rehearsals, coding sessions, both, or neither. Let $D$ be the event that a student attends drama rehearsals and $C$ be the event that a student attends coding sessions. It is known that $39$ students attend drama rehearsals and $25$ students attend neither activity. The Venn diagram represents the situation, where $x$ is the number of students who attend both activities.

Two fair four-sided dice are rolled. Each die is labelled $1$, $2$, $3$, $4$. Let $S$ be the event that the sum of the two numbers is prime. Let $T$ be the event that the product of the two numbers is even.

A security code is generated by first selecting a number $x$ from $\\{1,2,3,4,5\\}$ and then selecting a number $y$ from $\\{1,2,3,4\\}$. All $20$ ordered pairs $(x,y)$ are equally likely.

Let $A$ be the event that the product $xy$ is even. Let $B$ be the event that $x+y\ge 6$.

A survey of $240$ commuters records whether they use a bicycle, event $B$, and whether they use a train, event $T$, during a typical week. It is known that $108$ commuters use a bicycle, $96$ use a train, and $P(B\mid T)=0.375$.

A workshop uses two checks on each product. Let $V$ be the event that a product fails the visual check and $E$ the event that it fails the electronic check. For one product, $P(V)=0.18$, $P(E)=0.11$ and $P(V\cup E)=0.25$.

A transport company simulated $1200$ bus journeys using its timetable model. Of these journeys, $520$ were during peak times and $680$ were off-peak. There were $91$ late peak-time journeys and $136$ late off-peak journeys.

Let $P$ be the event that a journey is during peak time, and let $L$ be the event that a journey is late.

A veterinary clinic classifies cats with a particular symptom into three groups: healthy $H$, mildly ill $M$, and seriously ill $S$. The prior probabilities are $\mathrm{P}(H)=0.10$, $\mathrm{P}(M)=0.25$ and $\mathrm{P}(S)=0.65$. A diagnostic test is positive with probabilities $0.85$, $0.30$ and $k$ for cats in groups $H$, $M$ and $S$ respectively. It is known that, given a positive test, the probability that a cat is healthy is $0.40$.

Songs played by a music app are classified as pop, rock or jazz. The probabilities that a randomly selected song is pop, rock or jazz are $0.50$, $0.30$ and $0.20$ respectively. The probability that a user skips a pop song within $10$ seconds is $p$. The corresponding probabilities for a rock song and a jazz song are $0.24$ and $0.09$ respectively. Given that a song is skipped within $10$ seconds, the probability that it is jazz is $0.12$.

A biased coin is tossed repeatedly. The probability of obtaining heads on any toss is $p$, where $0<p<1$. Tosses are independent.

Ten cards are numbered $1$ to $10$. Two cards are selected at random, one after another, without replacement. Let $E$ be the event that the sum of the two numbers is odd. Let $F$ be the event that the product of the two numbers is a multiple of $3$.

Events $A$ and $B$ are in the same sample space. It is given that $P(A)=\frac{2}{5}$, $P(B\mid A)=\frac{1}{4}$ and $P(B\mid A')=k$.

A mineral sample is known to have come from exactly one of three sites, $A$, $B$ and $C$. The probabilities that it came from sites $A$, $B$ and $C$ are $\frac{2}{5}$, $\frac{2}{5}$ and $\frac{1}{5}$ respectively. A laboratory test records whether the sample contains a particular marker, $M$. The probabilities of recording the marker for samples from sites $A$, $B$ and $C$ are $\frac{1}{6}$, $\frac{1}{3}$ and $\frac{3}{4}$ respectively.

A message received by a company is either internal, $I$, or external, $E$. The probability that a message is internal is $\frac{3}{5}$. An automatic filter flags a message. Let $F$ be the event that a message is flagged. It is known that $P(F\mid I)=p$ and $P(F\mid E)=\frac{1}{4}$.

A box contains $6$ red tokens, $5$ blue tokens and $4$ yellow tokens. Three tokens are selected at random without replacement.

Let $R$ be the event that exactly two red tokens are selected. Let $Y$ be the event that at least one yellow token is selected.

A box contains $9$ adult wristbands, $6$ child wristbands and $5$ VIP wristbands. Two wristbands are selected at random without replacement.

A wildlife camera records one animal at a time. The animal is classified as deer, fox or badger with probabilities $0.55$, $0.30$ and $0.15$ respectively. The probability that the motion alarm is triggered is $0.20$ for a deer, $0.45$ for a fox and $0.70$ for a badger.

Let $A$ be the event that the alarm is triggered.

Emails received by a company are classified as personal, work-related or advertising. The probabilities of these classifications are $0.25$, $0.50$ and $0.25$ respectively. The probability that an email is marked urgent is $0.08$ if it is personal, $p$ if it is work-related, and $0.02$ if it is advertising.

It is given that, for an email marked urgent, the probability that it is work-related is $0.80$.