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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.
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
B. Twelve fair dice are tossed independently and at least two "6"s appear. C. Eighteen fair dice are tossed independently and at least three "6"s appear. [3] Pepys initially thought that outcome C had the highest probability, but Newton correctly concluded that outcome A actually has the highest probability.
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]
Candès and Recht [3] proved that with assumptions on the sampling of the observed entries and sufficiently many sampled entries this problem has a unique solution with high probability. An equivalent formulation, given that the matrix M {\displaystyle M} to be recovered is known to be of rank r {\displaystyle r} , is to solve for X ...
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
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 ...
Its solution is whichever combination of taxis and customers results in the least total cost. Now, suppose that there are four taxis available, but still only three customers. This is an unbalanced assignment problem. One way to solve it is to invent a fourth dummy task, perhaps called "sitting still doing nothing", with a cost of 0 for the ...