 | This article includes a list of general references, but it lacks sufficient corresponding inline citations .(February 2015) |
The necktie paradox is a puzzle and paradox with a subjective interpretation of probability theory describing a paradoxical bet advantageous to both involved parties. The two-envelope paradox is a variation of the necktie paradox.
Two persons, each given a necktie, start arguing over who has the cheaper one. The person with the more expensive necktie must give it to the other person.
The first person reasons as follows: winning and losing are equally likely. If I lose, then I will lose the value of my necktie. But if I win, then I will win more than the value of my necktie. Therefore, the wager is to my advantage. The second person can consider the wager in exactly the same way; thus, paradoxically, it seems both persons have the advantage in the bet.
The paradox can be resolved by giving more careful consideration to what is lost in one scenario ("the value of my necktie") and what is won in the other ("more than the value of my necktie"). If one assumes for simplicity that the only possible necktie prices are $20 and $40, and that a person has equal chances of having a $20 or $40 necktie, then four outcomes (all equally likely) are possible:
| Price of 1st person's tie | Price of 2nd person's tie | 1st person's gain/loss |
|---|---|---|
| $20 | $20 | 0 |
| $20 | $40 | Gain $40 |
| $40 | $20 | Lose $40 |
| $40 | $40 | 0 |
The first person has a 50% chance of a neutral outcome, a 25% chance of gaining a necktie worth $40, and a 25% chance of losing a necktie worth $40. Turning to the losing and winning scenarios: if the person loses $40, then it is true that they have lost the value of their necktie; and if they gain $40, then it is true that they have gained more than the value of their necktie. The win and the loss are equally likely, but what we call "the value of the necktie" in the losing scenario is the same amount as what we call "more than the value of the necktie" in the winning scenario. Accordingly, neither person has the advantage in the wager. [1]
This paradox is a rephrasing of the simplest case of the two envelopes problem, and the explanation of the resolution is essentially the same.
Games available in most casinos are commonly called casino games. In a casino game, the players gamble cash or casino chips on various possible random outcomes or combinations of outcomes. Casino games are also available in online casinos, where permitted by law. Casino games can also be played outside of casinos for entertainment purposes, like in parties or in school competitions, on machines that simulate gambling.
The gambler's fallacy, also known as the Monte Carlo fallacy or the fallacy of the maturity of chances, is the belief that, if an event has occurred less frequently than expected, it is more likely to happen again in the future. The fallacy is commonly associated with gambling, where it may be believed, for example, that the next dice roll is more than usually likely to be six because there have recently been fewer than the expected number of sixes.
Roulette is a casino game which was likely developed from the Italian game Biribi. In the game, a player may choose to place a bet on a single number, various groupings of numbers, the color red or black, whether the number is odd or even, or if the number is high or low.
Spread betting is any of various types of wagering on the outcome of an event where the pay-off is based on the accuracy of the wager, rather than a simple "win or lose" outcome, such as fixed-odds betting or parimutuel betting.
Fixed-odds betting is a form of gambling where individuals place bets on the outcome of an event, such as sports matches or horse races, at predetermined odds. In fixed-odds betting, the odds are fixed and determined at the time of placing the bet. These odds reflect the likelihood of a particular outcome occurring. If the bettor's prediction is correct, they receive a payout based on the fixed odds. This means that the potential winnings are known at the time of placing the bet, regardless of any changes in the odds leading up to the event.
In probability theory, odds provide a measure of the likelihood of a particular outcome. When specific events are equally likely, odds are calculated as the ratio of the number of events that produce that outcome to the number that do not. Odds are commonly used in gambling and statistics.
Pascal's wager is a philosophical argument advanced by Blaise Pascal (1623–1662), seventeenth-century French mathematician, philosopher, physicist, and theologian. This argument posits that individuals essentially engage in a life-defining gamble regarding the belief in the existence of God.
A martingale is a class of betting strategies that originated from and were popular in 18th-century France. The simplest of these strategies was designed for a game in which the gambler wins the stake if a coin comes up heads and loses if it comes up tails. The strategy had the gambler double the bet after every loss, so that the first win would recover all previous losses plus win a profit equal to the original stake. Thus the strategy is an instantiation of the St. Petersburg paradox.
In a thought experiment proposed by the Italian probabilist Bruno de Finetti in order to justify Bayesian probability, an array of wagers is coherent if it does not result in a Dutch book.
The St. Petersburg paradox or St. Petersburg lottery is a paradox involving the game of flipping a coin where the expected payoff of the theoretical lottery game approaches infinity but nevertheless seems to be worth only a very small amount to the participants. The St. Petersburg paradox is a situation where a naïve decision criterion that takes only the expected value into account predicts a course of action that presumably no actual person would be willing to take. Several resolutions to the paradox have been proposed, including the impossible amount of money a casino would need to continue the game indefinitely.
Vigorish is the fee charged by a bookmaker for accepting a gambler's wager. In American English, it can also refer to the interest owed a loanshark in consideration for credit. The term came to English usage via Yiddish slang which was itself a loanword from Russian.
In decision theory, the Ellsberg paradox is a paradox in which people's decisions are inconsistent with subjective expected utility theory. John Maynard Keynes published a version of the paradox in 1921. Daniel Ellsberg popularized the paradox in his 1961 paper, "Risk, Ambiguity, and the Savage Axioms". It is generally taken to be evidence of ambiguity aversion, in which a person tends to prefer choices with quantifiable risks over those with unknown, incalculable risks.
The two envelopes problem, also known as the exchange paradox, is a paradox in probability theory. It is of special interest in decision theory and for the Bayesian interpretation of probability theory. It is a variant of an older problem known as the necktie paradox. The problem is typically introduced by formulating a hypothetical challenge like the following example:
Imagine you are given two identical envelopes, each containing money. One contains twice as much as the other. You may pick one envelope and keep the money it contains. Having chosen an envelope at will, but before inspecting it, you are given the chance to switch envelopes. Should you switch?
In probability theory, the Kelly criterion is a formula for sizing a bet. The Kelly bet size is found by maximizing the expected value of the logarithm of wealth, which is equivalent to maximizing the expected geometric growth rate. Assuming that the expected returns are known, the Kelly criterion leads to higher wealth than any other strategy in the long run. John Larry Kelly Jr., a researcher at Bell Labs, described the criterion in 1956.
Asian handicap betting is a form of betting on football in which teams are handicapped according to their form so that a stronger team must win by more goals for a bet on them to win. The system originated in Indonesia and gained popularity in the early 21st century. It is a form of spread betting. Handicaps typically range from one-quarter goal to several goals, in increments of half- or even quarter-goals.
The Monty Hall problem is a brain teaser, in the form of a probability puzzle, based nominally on the American television game show Let's Make a Deal and named after its original host, Monty Hall. The problem was originally posed in a letter by Steve Selvin to the American Statistician in 1975. It became famous as a question from reader Craig F. Whitaker's letter quoted in Marilyn vos Savant's "Ask Marilyn" column in Parade magazine in 1990:
Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?
The mathematics of gambling is a collection of probability applications encountered in games of chance and can get included in game theory. From a mathematical point of view, the games of chance are experiments generating various types of aleatory events, and it is possible to calculate by using the properties of probability on a finite space of possibilities.
In gambling parlance, making a book is the practice of laying bets on the various possible outcomes of a single event. The phrase originates from the practice of recording such wagers in a hard-bound ledger and gives the English language the term bookmaker for the person laying the bets and thus 'making the book'.
In probability theory, Proebsting's paradox is an argument that appears to show that the Kelly criterion can lead to ruin. Although it can be resolved mathematically, it raises some interesting issues about the practical application of Kelly, especially in investing. It was named and first discussed by Edward O. Thorp in 2008. The paradox was named for Todd Proebsting, its creator.
Siegel's paradox is the phenomenon that uncertainty about future prices can theoretically push rational consumers to temporarily trade away their preferred consumption goods for non-preferred goods, as part of a plan to trade back to the preferred consumption goods after prices become clearer. For example, in some models, Americans can expect to earn more American dollars on average by investing in Euros, while Europeans can expect to earn more Euros on average by investing in American dollars. The paradox was identified by economist Jeremy Siegel in 1972.