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The birthday problem in this more generic sense applies to hash functions: the expected number of N-bit hashes that can be generated before getting a collision is not 2 N, but rather only 2 N ⁄ 2. This is exploited by birthday attacks on cryptographic hash functions and is the reason why a small number of collisions in a hash table are, for ...
The latter proposition may remain valid, even if it is certain that some instances of the generalisation are false. It is more likely than not, for example, that any number will be divisible either by two or by three, but it is not more likely than not that all numbers are divisible either by two or by three.
Suppose that we are asked to bet for or against the outcome "6". We might reason that there are two relevant outcomes here "6" or "not 6", and that these are mutually exclusive and exhaustive. A common fallacy is assigning the probability 1/2 to each of the two outcomes, when "not 6" is five times more likely than "6."
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 (whose occurrences are independent and identically distributed) has occurred less frequently than expected, it is more likely to happen again in the future (or vice versa).
The second of five storms that will slam the eastern half of the United States with snow and ice over a two week period is on the way – and this one has more snow than the first. Not 1. Not 2.
The standard is met if the proposition is more likely to be true than not true. Lord Denning, in Miller v. Minister of Pensions, [15] described it simply as "more probable than not". Another high-level way of interpreting that is that the plaintiff's case (evidence) be 51% likely.
[1] [2] It should not be confused with risk aversion, which describes the rational behavior of valuing an uncertain outcome at less than its expected value. When defined in terms of the pseudo-utility function as in cumulative prospect theory (CPT), the left-hand of the function increases much more steeply than gains, thus being more "painful ...
The departure of the upper tail of the distribution from the expected trend along the diagonal is due to the presence of substantially more large test statistic values than would be expected if all null hypotheses were true. The red point corresponds to the fourth largest observed test statistic, which is 3.13, versus an expected value of 2.06.