A Markov chain has states $A$, $B$ and $C$, with transition matrix

The system starts in state $A$.

Each day, a commuter travels by car, bus or bicycle. If the commuter travels by car, the probabilities of using car, bus and bicycle the next day are $0.70$, $0.20$ and $0.10$, respectively. From bus, the corresponding probabilities are $0.25$, $0.60$ and $0.15$. From bicycle, they are $0.40$, $0.10$ and $0.50$.

A wildlife reserve classifies an animal as being either inside a protected zone, $I$, or outside it, $O$. The monthly transition matrix is

where the state order is $I,O$.

The weekly movement of customers between three supermarkets, $A$, $B$ and $C$, is modelled by

The chain is regular.

A Markov chain has transition matrix

A two-state Markov chain has transition matrix

Its steady-state probability matrix is

A research task can be active, completed or abandoned. At the end of each week, an active task remains active with probability $0.60$, is completed with probability $0.30$, and is abandoned with probability $0.10$. Once completed or abandoned, its state does not change.

At an outdoor market, the probability that it rains is $0.30$. If it rains, the probability that a stallholder arrives late is $0.55$. If it does not rain, the probability that a stallholder arrives late is $0.12$.

Let $R$ denote the event that it rains and $L$ the event that a stallholder arrives late.

A survey of $240$ residents records whether they use a community swimming pool, $S$, and whether they use a community gym, $G$. There are $138$ residents who use the pool, $112$ who use the gym and $64$ who use both.

A game has a theoretical probability of success of $0.25$. Three computer simulations of the game produce the results shown.

A plant produces seeds that are classified as round, $R$, or non-round, $R'$. A seed may also germinate, $G$. For a randomly selected seed, $P(R)=0.55$, $P(G\mid R)=0.72$ and $P(G\mid R')=0.38$.

At an international conference, $62\%$ of delegates attend an artificial intelligence session, $A$, and $48\%$ attend a sustainability session, $B$. A total of $31\%$ attend both sessions.

A container holds $6$ gold tokens, $5$ silver tokens and $4$ black tokens. Three tokens are selected at random without replacement.

A fair four-sided die is numbered $1$ to $4$ and a fair eight-sided die is numbered $1$ to $8$. Both dice are rolled once. Let $A$ be the event that the sum is greater than $8$, and let $B$ be the event that the product is a multiple of $6$.

A screening test is used for a condition affecting $4\%$ of a population. For a person with the condition, the probability of a positive result is $0.92$. For a person without the condition, the probability of a positive result is $0.07$.

Let $C$ denote having the condition and $T$ denote receiving a positive test result.

For a flight selected at random, let $D$ be the event that the flight is delayed and $B$ the event that its passengers experience a baggage-handling problem. It is known that

For a particular plant species, the probability that a seed produces a red-flowered plant is $0.35$. The probability that a plant is tall is $0.72$ if it has red flowers and $0.48$ if it does not have red flowers.

Let $R$ denote red flowers and $T$ denote a tall plant.

A library classifies each book as on a shelf, $S$, on loan, $L$, or being reshelved, $R$. Daily movement is modelled by the transition matrix

where state matrices are column matrices in the order $S,L,R$.

A wetland is classified each year as healthy, $H$, or degraded, $D$. Its condition is modelled by

where the state order is $H,D$. Initially, the probabilities that the wetland is healthy and degraded are $0.90$ and $0.10$, respectively.

A parcel is classified as in transit, $T$, delivered, $D$, or lost, $L$. Each day, a parcel in transit remains in transit with probability $0.58$, is delivered with probability $0.37$, and is lost with probability $0.05$. Delivered and lost are absorbing states. The state order is $T,D,L$.

Tourists in a country are classified as staying in a coastal region, $C$, a mountain region, $M$, or a lake region, $L$. Their weekly movement is modelled by

Initially, the state probability matrix is

A coastal warning system issues an alert after severe rainfall with probability $0.91$. The probability of severe rainfall on a particular day is $0.12$. On a day without severe rainfall, the system issues a false alert with probability $0.03$. A simulation of $10\,000$ days produced $1400$ alerts.

A container holds $6$ amber beads, $5$ blue beads and $4$ clear beads. Three beads are selected at random.

In a game, a fair eight-sector spinner numbered $1$ to $8$ is spun and a coin is tossed. The coin toss and spinner result are independent, both for the fair coin and for the biased coin in part b. A player wins if the spinner number is divisible by $3$, or if the coin shows heads and the spinner number is odd. The word “or” is inclusive.

A curator has $12$ visually identical keys, of which $4$ open a secure cabinet. Keys are selected at random.

An animal is classified each month as occupying forest, $F$, edge habitat, $E$, or open land, $O$. From $F$, the next-month probabilities for $F,E,O$ are $0.70,0.20,0.10$. From $E$, they are $0.25,0.60,0.15$. From $O$, they are $0.15,0.35,0.50$. State matrices use the order $F,E,O$.

Daily weather at an observatory is modelled as sunny, $S$, or cloudy, $C$. After a sunny day, the next day is sunny with probability $0.75$. After a cloudy day, the next day is sunny with probability $0.40$. Let $x_n$ be the probability of sunny weather after $n$ transitions, with $x_0=1$.

In a game, a player's score is one of $0,1,2,3,4$. From scores $1$, $2$ or $3$, the score increases by one with probability $0.55$ and decreases by one with probability $0.45$. Scores $0$ and $4$ are absorbing states. State matrices use the order $0,1,2,3,4$.

Each day, a patient is classified as stable, $S$, in intensive care, $I$, or discharged, $D$. A stable patient remains stable with probability $0.70$, moves to intensive care with probability $0.20$, and is discharged with probability $0.10$. A patient in intensive care moves to stable with probability $0.25$, remains in intensive care with probability $0.60$, and is discharged with probability $0.15$. Discharged is an absorbing state.

Customers of an energy company have either a fixed-price contract, $F$, or a variable-price contract, $V$. Each month, a proportion $p$ of fixed-price customers changes to a variable-price contract, while $12\%$ of variable-price customers changes to a fixed-price contract. The steady-state probability matrix is

where the state order is $F,V$.

A screening test is used for a condition affecting $3.5\%$ of a population. For a person with the condition, the probability of a positive result is $0.94$. For a person without the condition, the probability of a negative result is $0.89$. Test results are assumed independent conditional on whether the person has the condition.

A communication station contains two components, $A$ and $B$. On a storm day, which occurs with probability $0.08$, the failure probabilities are $0.30$ and $0.20$, respectively. On a calm day, the corresponding probabilities are $0.04$ and $0.03$. Component failures are independent conditional on the type of day.