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In probability theory, odds provide a measure of the probability of a particular outcome. Odds are commonly used in gambling and statistics.For example for an event that is 40% probable, one could say that the odds are "2 in 5", "2 to 3 in favor", or "3 to 2 against".
Decimal odds are a single value, greater than 1, representing the amount to be paid out for each unit bet. For example, a bet of £40 at 6 − 4 (fractional odds) will pay out £40 + £60 = £100. The equivalent decimal odds are 2.5; £40 × 2.5 = £100. We can convert fractional to decimal odds by the formula D = (b + a) ⁄ b.
An alternative method of calculating the odds is to note that the probability of the first ball corresponding to one of the six chosen is 6/49; the probability of the second ball corresponding to one of the remaining five chosen is 5/48; and so on. This yields a final formula of
Stated in terms of odds, Bayes' rule states that the posterior odds of two alternatives, and , given an event , is the prior odds, times the likelihood ratio. As an equation: O ( A 1 : A 2 ∣ B ) = O ( A 1 : A 2 ) ⋅ Λ ( A 1 : A 2 ∣ B ) . {\displaystyle O(A_{1}:A_{2}\mid B)=O(A_{1}:A_{2})\cdot \Lambda (A_{1}:A_{2}\mid ...
Probability of being dealt a four of a kind in poker 10 −3: Milli-(m) 1.3×10 −3: Gaussian distribution: probability of a value being more than 3 standard deviations from the mean on a specific side [17] 1.4×10 −3: Probability of a human birth giving triplets or higher-order multiples [18] Probability of being dealt a full house in poker ...
The odds strategy is optimal, that is, it maximizes the probability of stopping on the last 1. The win probability of the odds strategy equals w = Q s R s {\displaystyle w=Q_{s}R_{s}} If R s ≥ 1 {\displaystyle R_{s}\geq 1} , the win probability w {\displaystyle w} is always at least 1/ e = 0.367879... , and this lower bound is best possible .
The probability is sometimes written to distinguish it from other functions and measure P to avoid having to define "P is a probability" and () is short for ({: ()}), where is the event space, is a random variable that is a function of (i.e., it depends upon ), and is some outcome of interest within the domain specified by (say, a particular ...
Probability density functions of the order statistics for a sample of size n = 5 from an exponential distribution with unit scale parameter. In statistics, the kth order statistic of a statistical sample is equal to its kth-smallest value. [1]