Games of Chance and Decisions
At the school fair there is a booth with this deal: you pay $\$10$ to choose a number from 1 to 6 and roll a die; if you guess right, you get back $\$50$. Is it worth playing many times? Probability lets you find out before you spend your Sunday.
Fair games
To find out whether it is fair, first count the possible cases and the cases favorable to each player, as in Laplace's rule: $P = \dfrac{\text{favorable cases}}{\text{possible cases}}$.
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 color 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 color and win $1$ point for a correct guess. Is it a fair game? Why?
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.
When the game isn't fair: how do we fix it?
If one player has a higher probability, we can compensate with points or bets: the player with the lower probability gets a bigger prize.
Counting with a tree diagram
If the game has several tokens or rolls, a tree diagram or a table of pairs helps you list all the cases before counting. For example, with two dice whose faces are prime numbers, make a $6 \times 6$ table with the products and count how many cells give the result you want.
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,…
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?
Two friends are playing a game drawing a card from a Spanish deck of cards: you win if a face card is drawn and your friend wins if a card from the suit of bastos is…
Two friends are playing a game drawing a card from a Spanish deck 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.
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…
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 color; Player $2$ wins if all three colors 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.
Carla and Jesús play with two dice whose faces are the first six prime numbers ($2$, $3$, $5$, $7$, $11$, $13$). Jesús proposes: "If the product is $49$, you win; if it…
Carla and Jesús play with two dice whose faces are the first six prime numbers ($2$, $3$, $5$, $7$, $11$, $13$). Jesús proposes: "If the product is $49$, you win; if it is $6$, I win." Carla does not accept.
a) How many pairs give a product of $49$? And a product of $6$?
b) Why do you think Carla does not accept? Calculate the probabilities.
c) Propose a product for each player that makes the game fair.
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?…
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.
Expected gain: is it worth playing?
When winning and losing involve different amounts, looking at the probability is not enough. You have to see how much you win or lose on average if you play many times.
If the probability of winning is $p$, in $N$ rounds you expect to win about $pN$ times. That is what is expected on average; it is not a guarantee for a short series.
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…
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?
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 carnival, a booth offers this game: you bet $\$1$ on a number from $1$ to $6$ and a die is rolled; if your number comes up, you get $\$5$ back (your pesos plus four…
At a carnival, a booth offers this game: you bet $\$1$ on a number from $1$ to $6$ and a die is rolled; if your number comes up, you get $\$5$ back (your pesos plus four more); otherwise, you lose your pesos.
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?
Several prizes: the expected value
When there are more than two outcomes, you do the same thing: multiply each prize by its probability and add. For $N$ attempts, the expected number of times for each outcome is $N$ times its probability.
At the fair booth shown in the figure, spinning the wheel costs $\$40…
At the fair booth shown in the figure, spinning the wheel costs $\$40$.
a) Calculate the probability of each outcome (no prize, bracelet, and stuffed animal).
b) If you play $100$ times, costing $\$4\,000$, what total value of prizes would you expect to receive?
c) Is it worth playing? Who does the game favor?
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…
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.
Chance has no memory
If a coin landed heads $6$ times in a row, what comes up now? Each flip is independent: the coin doesn't remember. The probability of heads is still $\tfrac{1}{2}$.
The same goes for the lottery: all numbers have the same probability of coming up, and the city where you buy your ticket or the endings that came up before change nothing.
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.
Spinners and dice that spring surprises
To compare two spinners or two dice, make a table with all the pairs of outcomes and count in how many each one wins. With 3 faces per die (or 3 outcomes per spinner), there are $9$ pairs.
A curious result: there can be three dice where the first beats the second, the second beats the third, and the third beats the first. They are called nontransitive.
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…
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…
Three special dice have these faces:
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Red die: $4$, $4$, $4$, $4$, $4$, $1$
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Green die: $2$, $2$, $2$, $5$, $5$, $5$
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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.