I recently stumbled across a fascinating puzzle from Bernoulli's Ars Conjectandi, and then found that part 3 of his book consists of 24 equally exciting worked problems. But I haven't found online the presentation of these problems.
See either Anders Hald's A History of Probability and Statistics and Their Applications before 1750 (2005), especially chapter 15, section 5...or Edith Dudley Sylla's translation The Art of Conjecturing, Together with Letter to a Friend on Sets in Court Tennis (2006).
If I misrepresented any of the problems, leave me a comment! I was rather quick and rushed in assembling the exercises, and I easily could have made mistakes. Also, I am fully aware some of these problems are ambiguous. Try working out every variant you can think of. The whole point (at least, to me) is that these present variations on a theme.
Problem 1. There are two balls in an urn, a "winner" ball and a "loser" ball. There are three players. The first player draws a ball. If it's the "loser" ball, the first player returns it to the urn; and if it's the "winner" ball, the first player wins the game. Should the first player lose, the second player performs the same task, and wins only if drawing the "winner" ball. Should the first and second players both draw the "loser" ball, the third player draws a ball. If the third player draws the "loser" ball, the house wins the game. What is the expected winnings for each player (and the bank)?
Problem 2. A variant of problem 1, each player bets some amount. If no player draws the winning ball, then they divide up the bets equally. (Example: player 1 bets 4 coins, player 2 bets 2 coins, player 3 bets 1 coin; if they all draw the "loser" ball, then each player receives (4 + 2 + 1)/3 coins back.) What is the expected winnings for each player?
Problem 3. Consider a tournament (i.e., a sequence of 2-player games). Two players compete in a game, where there will be only one winner. The winner plays the game against player 3. The winner of the second match plays against player 4. And so on until player 6. If the lots of each player in the games are equal, how do the lots of the later players compare/increase over those of the earlier players?
Problem 4. As a variant of problem 3, the lot of the player who wins the first game is stipulated to be double that of the third player, who plays only in the second game, and so on. Compare the expected winnings for each player.
To be clear, Bernoulli means the relative probability mass of winning the first round are even for both players (each player has 1 favorable outcome for an odds of 1:1, or 50% probability of winning), but the relative probability mass for the winner of the first round to beat player 3 is doubled (the victor of the first match has twice the favorable outcomes as compared to game 1 [i.e., 2 favorable outcomes] whereas player 3 has 1 favorable producing a 2/3 probability favoring the victor of the first game to win the second), and the relative probability mass of the winner of the second round is doubled to determine the odds of winning the third game (so if the victor of the first game also won the second game, this victor has 4 favorable outcomes, but player 4 has 1 favorable outcome, the probability of winning this particular game for the victor of the first two games would be 4/5), and so on.
Problem 5. "This is the third problem of Huygens' Appendix." A wagers against B, that of 40 cards, of which 10 of each color, he will draw 4 of them in a way to have one of each color. (Or, for modern readers, consider a standard playing deck with Jacks, Queens, Kings discarded; A wages that B cannot draw one card from each suit.) What is the probability that A wins the bet? [Alleged solution: One finds in this case that the chance of A is to that of B as 1000 is to 8139.]
Problem 6. "This is the fourth problem of Huygens' Appendix." One takes 12 tokens of which 4 white and 8 black. A wagers against B that among 7 tokens that he will draw from them blindly, there will be found 3 white. One demands the ratio of the chance of A to that of B; i.e., what are the odds A wins? (There is some ambiguity in the historic text; namely, is it exactly 3 white or at least 3 white?)
Face Cards
Problem 7. Let there be a single face card in a pack of n cards, and the first player (of m players) to draw it wins. If no one draws it, and cards remain, then the players continue drawing cards. What is the probability of each player to win? What if n = mk (the number of players divides the number of cards)?
Problem 8. As a variant of Problem 7, what if there were j face cards in the deck and the winner is still the first person to draw a face card? What is the probability for each player to win?
Problem 9. As a variant of Problem 8, what if the players keep drawing cards until the deck is exhausted, and the winner is the player with the most face cards drawn? For ties, the winnings are split among the winners. (Assume each player pays 1 coin to play, for example.) What is the expected winnings for each player?
Problem 10. As an extension to Problem 9, what if we permit any player to sell their position? Specifically Bernoulli considers four players (A, B, C, D) with a deck of 36 cards, of which 16 are face cards. Each player receives cards in rounds until 23 cards have been distributed, A has received 4 face cards, B has 3, C received 2, and D a modest 1 face card, so that there remain 13 cards among which there are 6 face cards. The fourth player D (who is next to receive a card), "seeing that almost all hope of his winning has vanished", wishes to sell his right to one of the others. How much should he sell it for and what are the expectations of the individual players?
Dice Puzzles
Problem 11. Throw a die. Then throw a second die. If they differ, the player wins a point; if they agree, the player loses a point. Then the player throws a third die. If it agrees with any previous die, lose one additional point for each agreement; otherwise, the player wins an additional point added to their running score. Do this for a total of 6 dice. What is the expected score for the player?
Problem 12. Similar to Problem 11, but the dice must be thrown in numerical order. E.g., the first die must be "1" for the player to get a point, otherwise the player loses a point; the second die must be "2" for the player to get a point, otherwise the player loses a point; and so on. What is the expected score for the player?
Problem 13. Three players (A, B, C) have a list of 6 numerals ("1", "2", ..., "6") on a sheet of paper before them. Each of them take turns round-robin. On a player's turn, they roll a die, and if the result is on their sheet of paper, then the player gets to eliminate it from his sheet (scratch off the number) and roll again; but if the player has already eliminated that number, then the next player gets the die. This process continues until someone eliminates all their written numerals. It happens, however, after a while that A has 2 numerals before him, B has 4, and C has 3; it's A's turn to throw. What are probabilities for each player to win? [Bernoulli notes This problem requires more labor and patience than ingenuity.
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Problem 14. There are k players. A given player throws a die, which shows its face to be m. This tells the player to throw m dice and sum the values shown. (We may choose m to be added to the score or not.) The player with the most points wins.
Here's a twist, though: one player may opt to beat a fixed number t of points. This value t is fixed by the rules of the game.
What's the expected winnings for each player if no one opts for the fixed points route? What's the expected winnings for each player if someone chooses to beat a fixed number of points?
Although Bernoulli didn't pitch it, what if there's a "bidding war" competition to determine which player may opt to beat a fixed number of points? Instead of having t be fixed, when one player asks to be the one to beat a fixed number of points, that player must offer a bid ("I want to beat x points"). Each player may pass (and no longer participates in the bidding anymore), or offers a higher bid ("I want to beat y > x points"). This continues until the maximum value of 35 is bid. What strategy works best in this bidding strategy?
Problem 15. As a variant of Problem 14, what if the fixed number of points is the square of the first toss of the die?
Problem 16 (Cinq et Neuf). This is a prototype of craps. Player 1 tosses a pair of dice. Player 1 wins if on his first toss she gets a 3, an 11, or any pair. But player 2 wins if player 1 tosses a 5 or 9.
If player 1's first toss is a 4, 6, 7, 8, or 10, then the game continues (player 1 keeps tossing a pair of dice) until either (a) a 5 or 9 appears [player 2 wins], or (b) player 1 tosses the first value [player 1 wins].
What is the probability of player 1 winning? (Player 2's probability of winning is, by definition, the complement of the probability for player 1 winning.)
Although not asked, what is the expected number of tosses for a given game?
Wheel Games
Problem 17. The player pays 4 coins to toss 4 balls on a roulette-like wheel. The roulette wheel has 32 pockets, each with labels "1", "2", ..., "8". There are four pockets labelled "1"; four pockets labelled "2"; and so on. Each pocket may contain at most 1 ball. The player wins the sum of the pockets's labels (for the pockets containing the balls). What is the expected winnings for the player? [Solution.]
Card Games
Problem 18 (Trijaques). We consider a "toy model" of poker.
We assemble a deck of 28 cards from a standard playing deck, by discarding the cards 2 through 8 for each suit. The values for each card are determined by its face value, but the Jack of Clubs and 9s are wild.
The player will be given 4 cards. The goal is for the player to assemble either a flush (a run of all 4 cards regardless of suit, e.g., "9, 10, J, Q"), or a pair, three of a kind, or four of a kind. The value of a hand is the sum of the value of the cards in the combination. The player with the highest valued hand wins the pot (or it is split among the highest valued hands).
But the sequence of play is as follows: each player is dealt 2 cards face down. Then the players bet. Then each player is dealt 2 cards face up.
What is the expected winnings for each player? What strategies could be considered in the betting process?
Problem 19. Consider a generic game, where one player is the "banker". The banker has an advantage over the other players (i.e., is more probable than any other player to win a given round). But the rules of the game may allow moving the banker role to another player.
Specifically, the banker has probability p of winning a round, and probability q of losing, with p + q = 1 and \(r = p - q > 0\). The banker has probability h of continuing the next round as banker, and probability k of losing the position as banker to another player, with \(t = h - k > 0\). Let a denote the amount won by either the player or the banker, whoever wins the round.
What is the expected winnings for the banker after "many" rounds?
Problem 20 (Capriludium, Bockspiel). At the beginning of the game, each player puts down their bet. Then the banker shuffles the deck, and divides it into equally sized hands. Each player (and the banker) gets a hand. Bernoulli says the player just turns the hand over without organizing it. If the punter [player who is not the banker] has their facing card be of equal or higher value compared to the banker, then the punter wins an amount of money equal to his bet/bid from the banker. Otherwise the player loses their wager to the banker. When the banker loses to all the players in a game, the next player becomes the banker.
After one round, the top cards are not yet discarded. New wagers are first made. Then the top card is discarded (collected by the banker for later shuffling).
If there are N = sf cards in the deck with s suits and f face cards (of value ranging between 1 to f), and suppose there are n players (including the banker) for n = 2, 3, 4.
What is the expected winnings for the banker? What is the expected number of rounds to be played for a given deck? How does it vary on the number of players? What is the probability the banker will remain in their role as banker after one hand? After h hands?
Problem 21 (Basset). The basic formalization is there are 2n cards, of which k are marked "a" and \(2n - k\) marked "b". For example, 2n = 52, and k = 4 (e.g., aces in a standard deck of playing cards). The player draws two cards in succession (no replacement). The possibly outcomes:
- ab = the banker wins 1 point
- ba = the player wins 1 point
- aa = the banker wins 1 point
- bb = toss the cards aside and draw 2 new cards (and consult this table of outcomes again)
What is the expected winnings for the banker after 1 hand? For 1 game (exhausting the whole deck)?
As a variant, we could try using the full rules for Basset, with the bizarre bet multiplying schemes.
Curious Puzzles
Problem 22. There are two players, Titus and Caius. Titus pays Caius one coin for each round where Titus will throw a single die. There are a possible outcomes, of which b favor Titus (Caius pays him one coin) and c favor Caius (where Titus wins nothing). If Titus throws one of the c cases continuously n times in a row, Caius must return all n coins to Titus. What is the expected winnings of Caius and Titus?
Problem 23 (Blinde Würffel, "Blind Dice"). We have 6 dice with a number on only one face, and blanks on the remaining faces. One die has "1" for its non-blank side, another has "2" for its non-blank side, and so on, so each label "1" through "6" may possibly show up. Blank sides are treated as having value 0. Suppose the player rolls all 6 dice, and wins the sum of the values shown. What is the player's expected winnings?
Problem 24. A variant of Problem 23, if the player gets no points, in 5 tosses in a row then the player gets his money back for those 5 tosses. What is the player's expected winnings now?
