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Alternatively, the probabilistic method can also be used to guarantee the existence of a desired element in a sample space with a value that is greater than or equal to the calculated expected value, since the non-existence of such element would imply every element in the sample space is less than the expected value, a contradiction.
These non-probabilistic existence theorems follow from probabilistic results: (a) a number chosen at random (uniformly on (0,1)) is normal almost surely (which follows easily from the strong law of large numbers); (b) some probabilistic inequalities behind the strong law. The existence of a normal number follows from (a) immediately.
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
To apply the method to a probabilistic proof, the randomly chosen object in the proof must be choosable by a random experiment that consists of a sequence of "small" random choices. Here is a trivial example to illustrate the principle. Lemma: It is possible to flip three coins so that the number of tails is at least 2. Probabilistic proof.
Akra–Bazzi method; Dynamic programming; Branch and bound; Birthday attack, birthday paradox; Floyd's cycle-finding algorithm; Reduction to linear algebra; Sparsity; Weight function; Minimax algorithm. Alpha–beta pruning; Probabilistic method; Sieve methods; Analytic combinatorics; Symbolic combinatorics; Combinatorial class; Exponential ...
then there is a nonzero probability that none of the events occurs. Lemma II (Lovász 1977; published by Joel Spencer [3]) If (+), where e = 2.718... is the base of natural logarithms, then there is a nonzero probability that none of the events occurs. Lemma II today is usually referred to as "Lovász local lemma".
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In statistics, the Bhattacharyya distance is a quantity which represents a notion of similarity between two probability distributions. [1] It is closely related to the Bhattacharyya coefficient, which is a measure of the amount of overlap between two statistical samples or populations.