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Then, with probability at least /, there is a unique set in that has the minimum weight among all sets of . It is remarkable that the lemma assumes nothing about the nature of the family F {\displaystyle {\mathcal {F}}} : for instance F {\displaystyle {\mathcal {F}}} may include all 2 n − 1 {\displaystyle 2^{n}-1} nonempty subsets.
If F is satisfiable, then with probability at least 1/4, some G i has a unique satisfying assignment. The idea of the reduction is to successively intersect the solution space of the formula F with n random linear hyperplanes in F 2 n {\displaystyle \mathbb {F} _{2}^{n}} .
In the simplest case, if one allocates balls into bins (with =) sequentially one by one, and for each ball one chooses random bins at each step and then allocates the ball into the least loaded of the selected bins (ties broken arbitrarily), then with high probability the maximum load is: [8]
Many probability text books and articles in the field of probability theory derive the conditional probability solution through a formal application of Bayes' theorem; among them books by Gill [51] and Henze. [52] Use of the odds form of Bayes' theorem, often called Bayes' rule, makes such a derivation more transparent. [34] [53]
Suppose l > t.In this case, integrating the joint probability density function, we obtain: = = (), where m(θ) is the minimum between l / 2 sinθ and t / 2 .. Thus, performing the above integration, we see that, when l > t, the probability that the needle will cross at least one line is
The solution to this particular problem is given by the binomial coefficient (+), which is the number of subsets of size k − 1 that can be formed from a set of size n + k − 1. If, for example, there are two balls and three bins, then the number of ways of placing the balls is ( 2 + 3 − 1 3 − 1 ) = ( 4 2 ) = 6 {\displaystyle {\tbinom {2 ...
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 PCP theorem states that NP = PCP[O(log n), O(1)],. where PCP[r(n), q(n)] is the class of problems for which a probabilistically checkable proof of a solution can be given, such that the proof can be checked in polynomial time using r(n) bits of randomness and by reading q(n) bits of the proof, correct proofs are always accepted, and incorrect proofs are rejected with probability at least 1/2.