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In probability theory and statistics, odds and similar ratios may be more natural or more convenient than probabilities. In some cases the log-odds are used, which is the logit of the probability. Most simply, odds are frequently multiplied or divided, and log converts multiplication to addition and division to subtractions.
Example of the optimal Kelly betting fraction, versus expected return of other fractional bets. In probability theory, the Kelly criterion (or Kelly strategy or Kelly bet) is a formula for sizing a sequence of bets by maximizing the long-term expected value of the logarithm of wealth, which is equivalent to maximizing the long-term expected geometric growth rate.
If the probability of being offered 5 to 1 odds is more than 50%, the Kelly bettor will actually make a negative bet at 2 to 1 odds (that is, bet on the 50/50 outcome with payout of 1/2 if he wins and paying 1 if he loses). In either case, his bet at 5 to 1 odds, if the opportunity is offered, is 40% minus 0.7 times his 2 to 1 bet.
Surprisal and evidence in bits, as logarithmic measures of probability and odds respectively. The logarithmic probability measure self-information or surprisal, [4] whose average is information entropy/uncertainty and whose average difference is KL-divergence, has applications to odds-analysis all by itself. Its two primary strengths are that ...
A betting strategy (also known as betting system) is a structured approach to gambling, in the attempt to produce a profit.To be successful, the system must change the house edge into a player advantage — which is impossible for pure games of probability with fixed odds, akin to a perpetual motion machine. [1]
They are exchange-traded markets established for trading bets in the outcome of various events. [1] The market prices can indicate what the crowd thinks the probability of the event is. A typical prediction market contract is set up to trade between 0 and 100%.
In this example, the probability of losing the entire bankroll and being unable to continue the martingale is equal to the probability of 6 consecutive losses: (10/19) 6 = 2.1256%. The probability of winning is equal to 1 minus the probability of losing 6 times: 1 − (10/19) 6 = 97.8744%. The expected amount won is (1 × 0.978744) = 0.978744.
In a discrete (i.e. finite state) market, the following hold: [2] The First Fundamental Theorem of Asset Pricing: A discrete market on a discrete probability space (,,) is arbitrage-free if, and only if, there exists at least one risk neutral probability measure that is equivalent to the original probability measure, P.