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In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes–no question, and each with its own Boolean-valued outcome: success (with probability p) or failure (with probability q = 1 − p).
In probability theory, Isserlis' theorem or Wick's probability theorem is a formula that allows one to compute higher-order moments of the multivariate normal distribution in terms of its covariance matrix. It is named after Leon Isserlis.
In probability theory and statistics, the hypergeometric distribution is a discrete probability distribution that describes the probability of successes (random draws for which the object drawn has a specified feature) in draws, without replacement, from a finite population of size that contains exactly objects with that feature, wherein each draw is either a success or a failure.
In probability theory, the coupon collector's problem refers to mathematical analysis of "collect all coupons and win" contests. It asks the following question: if each box of a given product (e.g., breakfast cereals) contains a coupon, and there are n different types of coupons, what is the probability that more than t boxes need to be bought ...
The probability of this happening is 1 in 13,983,816. The chance of winning can be demonstrated as follows: The first number drawn has a 1 in 49 chance of matching. When the draw comes to the second number, there are now only 48 balls left in the bag, because the balls are drawn without replacement .
In probability theory and statistics, the geometric distribution is either one of two discrete probability distributions: The probability distribution of the number X {\displaystyle X} of Bernoulli trials needed to get one success, supported on N = { 1 , 2 , 3 , … } {\displaystyle \mathbb {N} =\{1,2,3,\ldots \}} ;
If g is a general function, then the probability that g(X) is valued in a set of real numbers K equals the probability that X is valued in g −1 (K), which is given by (). Under various conditions on g , the change-of-variables formula for integration can be applied to relate this to an integral over K , and hence to identify the density of g ...
Log–log graph of the probability that a number starts with the digit(s) n, for a distribution satisfying Benford's law. The points show the exact formula, P(n) = log 10 (1 + 1/n). The graph tends towards the dashed asymptote passing through (1, log 10 e) with slope −1 in log–log scale. The example in yellow shows that the probability of a ...