Exercises: 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?

In a game, you win if you roll a $5$ or a $6$ when rolling a die, and you lose in any other case.

a) What is the probability of winning?

b) If you play $30$ times, how many wins would you expect?

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?

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?

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.

At your school's end-of-year festival, a charity stall is set up with an urn containing $15$ black balls and $5$ red balls, all of the same size and texture. Participating costs $\text{€}\ 1$ per draw (which is not refunded). If you draw a red ball, you receive a net prize of $\text{€}\ 2$ (that is, your $\text{€}\ 1$ coin is returned and you are given an additional $\text{€}\ 2$); if you draw a black ball, you receive nothing and lose the euro you bet.

a) Before doing any written mathematical calculations, estimate at a glance whether you think this game benefits the player or the charity stall in the long run. Briefly explain the intuition behind your estimate by comparing the proportion of balls and the prize amount.

b) Calculate the probability of winning ($P(\text{red})$) and the probability of losing ($P(\text{black})$) when drawing a ball.

c) Calculate the expected net gain per game for a player and the total expected balance if a person decided to play $60$ times in a row (returning the ball to the urn after each attempt).

d) Check whether your initial estimate from part a) matches the mathematical result obtained and argue whether it is a fair game or if there is fairness.

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.

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?

Three counters are flipped: one with green and red sides, another with green and blue, and another with red and blue. Player $1$ wins if any two counters show the same colour; Player $2$ wins if all three colours are different.

a) Using a tree diagram, calculate the probability of winning for each player.

b) Is it a fair game?

c) Propose a change in the rules or in the points system to make it a fair game.

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?

Aoife and Seán play with two dice whose faces are the first six prime numbers ($2$, $3$, $5$, $7$, $11$, $13$). Seán proposes: "If the product is $49$, you win; if it is $6$, I win." Aoife does not accept.

a) How many pairs give a product of $49$? And a product of $6$?

b) Why do you think Aoife does not accept? Calculate the probabilities.

c) Propose a product for each player that makes the game fair.

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.

Two friends are playing a game drawing a card from a Spanish pack of cards: you win if a face card is drawn and your friend wins if a card from the suit of bastos is drawn.

a) Calculate the probability of each person winning.

b) Who has the advantage?

c) If you want the game to be fair without changing the events, how many points should each person award for a win? Explain your proposal.

A ball is drawn from a bag containing $4$ red balls and $4$ green balls: if it is red, you win $3$ points; if it is green, your friend wins $5$ points. After each round, the ball is put back in the bag.

a) What is the probability of each person scoring?

b) If you play $40$ rounds, what score would you expect?

c) Is it a fair game? Why?

You are designing the rules for an adventure board game. A player reaches a secret door and can attempt to pick the lock by paying a fixed cost of $3$ gold coins. To resolve the attempt, a fair six-sided die numbered from $1$ to $6$ is rolled. If a multiple of $3$ ($3$ or $6$) is rolled, the attempt succeeds and the chest grants a loot of $P$ gold coins in net profit (in addition to getting back the $3$ coins bet). If any other outcome occurs ($1$, $2$, $4$, or $5$), the attempt fails and the player permanently loses their $3$ coins.

a) Calculate the probability of succeeding and the probability of failing on a roll.

b) Set up the equation for the expected net gain and state what value the net prize $P$ must have for the decision to pick the lock to be a mathematically fair game (expected gain equal to $0$).

c) Imagine that the rulebook is already printed and strictly sets the net prize at $P = 4$ coins and the bet at $3$ coins. The design team asks you to modify the die rule so that the game becomes fair simply by changing the number of winning faces. Is it possible to make the game fair by choosing a specific number of winning faces on a $6$-sided die? Mathematically prove why it is possible or impossible.

The community council of a small town in County Wicklow is organizing a hiking trip for young people. Equipment rental and transportation have a fixed cost of €40 per participant, which is paid in advance. According to historical records from the Met Éireann for that time of year, the probability of a weather alert being issued that forces the activity to be canceled for safety is $20\,\%$ ($0.20$). If the activity is canceled and you do not have insurance, $100\,\%$ of the money paid is lost.

An insurance company offers an optional policy for €10 per person: if the activity is canceled, the insurer fully reimburses the €40 cost of the activity.

a) Model the situation by calculating the expected financial balance (expected loss) for a person who decides not to purchase insurance.

b) If the person purchases insurance, they incur a certain, definitive cost of €10. Compare this amount with the expected loss you obtained in part a).

c) If $150$ young people participate in the trip and none of them purchase insurance, what total financial loss is the group of participants expected to suffer on average? How much would the group pay in total if everyone purchased the policy?

d) From a strictly mathematical expected balance perspective, who does the policy favor, the hikers or the insurance company? Explain what non-mathematical reasons might lead a family to purchase insurance despite this result.

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.

A spinner has $6$ equal sectors: $2$ red and $4$ blue. Two players bet: player A on red and player B on blue.

a) What are the probabilities of winning for each player?

b) If in each game the winner takes the points that both bet, suggest how many points each player should bet to make the game fair. Justify your proposal.

c) Design another division of sectors and bets that also makes a fair game.

In a video game, a chest costs $25$ gold coins and contains: a common item (worth $10$ coins) with probability $0.7$; a rare item (worth $50$ coins) with probability $0.25$; or an epic item (worth $300$ coins) with probability $0.05$.

a) If you buy $100$ chests, how many items of each type would you expect to get?

b) What would be the total expected value of these items? And the total expenditure?

c) Is it worth buying chests in the long run? Justify your answer.

At the local fair, there is an attraction where two standard six-sided dice (numbered from $1$ to $6$) are rolled simultaneously. Participating costs $\text{€}\ 2$, and the stallholder lets you choose between two betting modalities before rolling:

• Sum Modality: You win if the sum of the two dice is $7$ or $8$. The net prize is $\text{€}\ 3$ (your $\text{€}\ 2$ coin is returned and you receive an additional $\text{€}\ 3$).

• Doubles Modality: You win if both dice show the same number (doubles: $1$-$1$, $2$-$2$, etc.). The net prize is $\text{€}\ 8$ (your $\text{€}\ 2$ is returned and you receive an additional $\text{€}\ 8$).

Two friends debate how to decide which modality to choose to have a more favorable gaming session:

• Method 1 (probability criterion): Argues that you only need to calculate the winning probability of each modality and choose the one with the higher probability, because the main priority is to guess right.

• Method 2 (expected gain criterion): Argues that you must weight the probability of winning and losing by the amounts at stake, calculating the expected net gain per game.

a) Apply Method 1: determine the sample space of the $36$ equally likely outcomes and calculate the probability of winning in each modality. According to this method, which modality should be chosen?

b) Apply Method 2: calculate the expected net gain per game for the Sum Modality and for the Doubles Modality. According to this method, which modality is preferable for the player?

c) Compare both methods and justify which one is the correct one for making decisions in games of chance. Clearly explain the limitation or conceptual trap that Method 1 falls into.

d) Does either of the two modalities offer a real mathematical advantage over the stallholder (positive expected net gain)? Justify your answer.

A supermarket chain in Ireland introduces a loyalty promotion in its mobile app. For every purchase over €30, the customer receives an entry to choose between two reward options on their next purchase:

• Option A (Direct discount): A guaranteed, immediate discount of €0.75 is applied.

• Option B (Digital roulette): An electronic wheel divided into $10$ sectors of equal size and probability is spun. In $6$ of them, it lands on “Thank you for participating” (€0 prize), in $3$ of them you get a €1 discount coupon, and in $1$ of them you get a special “Super Savings” coupon of €$X$ net discount.

A customer comments on social media: “I always choose the Digital roulette because $4$ out of the $10$ sectors have a prize and, if you make $10$ purchases in a row, statistics guarantees you the Super Savings coupon of €$X$ at least once”.

a) Reason mathematically why the customer's claim on social media is false: does making $10$ spins guarantee that you will hit the Super Savings coupon sector at least once? Support your answer with the principle that chance has no memory.

b) Model the expected gain per spin of Option B by expressing it as a mathematical formula in terms of the unknown value $X$.

c) Set up an inequality and calculate the minimum value the special coupon prize $X$ must have for it to be mathematically more profitable in the long run for a consumer to choose Option B over Option A.

d) If the company sets $X = €5$ and a family makes $40$ purchases per year at that supermarket using the app, calculate the total expected cumulative savings in a year with each option. In light of the results, which option would you recommend to someone looking to maximize their annual savings?

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.