Games of chance and decisions
A fairground stall makes you an offer: you stake 1 € on a number from 1 to 6 and roll a die; if you guess correctly, you get 5 € back. Is it worth playing many times? Probability helps you find out before you spend a single euro.
Fair or equitable games
Three friends play a game by tossing two coins: Adrián wins if two heads come up, Blanca if two tails come up, and Carlos if one head and one tail come up. a) Write down…
Three friends play a game by tossing two coins: Adrián wins if two heads come up, Blanca if two tails come up, and Carlos if one head and one tail come up.
a) Write down the sample space.
b) Calculate the probability of each person winning.
c) Is it a fair game? Who has the advantage?
A spinner is divided into two equal halves, one red and one blue. Two players each bet on a colour and win $1$ point for a correct guess. Is it a fair game? Why?
A spinner is divided into two equal halves, one red and one blue. Two players each bet on a colour and win $1$ point for a correct guess. Is it a fair game? Why?
When the game is not equitable
Elsa and Bruno roll two dice: Elsa wins if the total is even and Bruno if it is odd. a) Is it a fair game? Calculate each person's probability of winning. b) What if,…
Elsa and Bruno roll two dice: Elsa wins if the total is even and Bruno if it is odd.
a) Is it a fair game? Calculate each person's probability of winning.
b) What if, instead of adding, they multiply the results and Elsa continues betting on even? Calculate the new probabilities and explain who has the advantage.
State whether these statements about lotteries and chance are true or false, giving reasons: a) Since the jackpot almost never lands in my town, I have a higher…
State whether these statements about lotteries and chance are true or false, giving reasons:
a) Since the jackpot almost never lands in my town, I have a higher probability of winning if I buy the ticket in another town.
b) The number $00\,027$ is less likely to be drawn than $23\,568$.
c) If the ending in $5$ has come up many times in recent draws, it is now less likely to come up.
d) If a coin has landed on heads $6$ times in a row, on the next toss tails is more likely.
Expected gain: is it worth playing?
When winning and losing hand out different points or amounts of money, it is not enough to look at who has the higher probability: you have to see how much you can expect to win or lose on average if you play many times.
In a game you bet $1$ counter on a number from $1$ to $6$ and roll a die: if your number comes up, you get $6$ counters (yours plus another $5$); if not, you lose your…
In a game you bet $1$ counter on a number from $1$ to $6$ and roll a die: if your number comes up, you get $6$ counters (yours plus another $5$); if not, you lose your counter.
a) What is the probability of winning?
b) If you play $60$ times, how many times would you expect to win? What would be your overall balance of counters?
c) Is the game fair? Why?
At a funfair, a stall offers this game: you bet $1\text{ €}$ on a number from $1$ to $6$ and a die is rolled; if your number comes up, you get $5\text{ €}$ back (your…
At a funfair, a stall offers this game: you bet $1\text{ €}$ on a number from $1$ to $6$ and a die is rolled; if your number comes up, you get $5\text{ €}$ back (your euro plus four more); otherwise, you lose your euro.
a) What is the probability of winning?
b) If you play $60$ times, what would your expected balance be?
c) Is it a fair game? If not, how much should the prize pay to make it fair?