Games of chance and decisions

Three friends play a game by tossing two coins: Seán wins if two heads come up, Aoife if two tails come up, and Cian 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?

Topic challenge. Is it ethical to set up a betting stall at the secondary school charity fair? You are going to design and audit a dice game to decide whether it is fair and profitable.

1. Sample space and Laplace: Design a game rolling two dice where you win if the sum is greater than or equal to $9$. Detail the sample space $E$ ($36$ equiprobable elementary events) and calculate the theoretical probability $P(\text{win})$ using Laplace's rule.

2. Experimental probability: Carry out $50$ real rolls at home (or simulate them on a spreadsheet using the random function). Record the results, calculate the relative frequency $f_r$ after each block of $10$ rolls, and analyse how it evolves with respect to the theoretical probability.

3. Real data and decision making: Check on the official website of the National Lottery the probability of matching $6$ numbers in the Lotto jackpot ($1$ in $13.98$ million, that is, $P \approx 0.000000071$). Write a reasoned proposal comparing both risks: set the price per game and the prize value of your game to raise funds transparently.

Assessment criteria:

• Rigorous determination of the sample space and exact application of Laplace's rule.

• Systematic collection of relative frequencies and critical comparison with the theoretical value.

• Economic proposal well-founded on data and comparison with real games of chance.

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 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.

Aoife and Cian are playing during break by tossing a fair €1 coin. They have agreed that if it lands on heads Aoife wins and if it lands on tails Cian wins, with the loser paying €2 to the winner on each toss. After four consecutive tosses in which it has landed on tails, Cian enthusiastically claims: “On the fifth toss I'm going to bet €5 instead of €2! Since four tails have come up in a row, by the law of chance it is practically impossible for tails to come up again; heads has to come up now to make up for the streak”.

a) Identify and clearly explain the reasoning error that Cian makes regarding how chance behaves.

b) What is the actual probability that heads comes up on the fifth toss? Has it changed compared to the first toss?

c) If Aoife were to accept that €5 bet for the fifth toss under the same rules, would that specific round be a fair game between the two? Justify your answer using the probabilities and the stakes involved.

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 $\text{€}\ 1$ on a number from $1$ to $6$ and a die is rolled; if your number comes up, you get $\text{€}\ 5$ 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?

At the funfair stall shown in the figure, spinning the wheel costs $\text{€}\ 2$.

a) Calculate the probability of each outcome (no prize, bracelet, and soft toy).

b) If you play $100$ times, costing $\text{€}\ 200$, what total value of prizes would you expect to receive?

c) Is it worth playing? Who does the game favour?

A game uses the three spinners shown in the figure. Each player chooses a spinner, both chosen spinners are spun, and whoever gets the higher number wins.

a) If the green spinner and the blue spinner face each other, who wins and with what probability?

b) If the blue spinner and the red spinner face each other, who wins and with what probability?

c) If the red spinner and the green spinner face each other, who wins and with what probability?

d) If your opponent chooses a spinner first, what strategy would you follow? Is there a spinner that is "the best"? Explain the result.

Three special dice have these faces:

• Red die: $4$, $4$, $4$, $4$, $4$, $1$

• Green die: $2$, $2$, $2$, $5$, $5$, $5$

• Blue die: $3$, $3$, $3$, $3$, $3$, $6$

Two players each choose a die and roll them; the higher number wins.

a) Calculate the probability that the red die beats the green die.

b) Calculate the probability that the green die beats the blue die.

c) Calculate the probability that the blue die beats the red die.

d) If your opponent chooses a die first, which die would you choose in each case? Does the result surprise you? Comment on it.