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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 mathematical analysis of the problem reveals that the expected number of trials needed grows as O (nlog (n)). For example, when n = 50 it takes about 225 samples to collect all 50 coupons.
The rule of division formulated in terms of functions: "If f is a function from A to B where A and B are finite sets, and that for every value y ∈ B there are exactly d values x ∈ A such that f (x) = y (in which case, we say that f is d-to-one), then |B| = |A|/d." [1]
The Bernoulli distribution, which takes value 1 with probability p and value 0 with probability q = 1 − p. The Rademacher distribution, which takes value 1 with probability 1/2 and value −1 with probability 1/2. The binomial distribution, which describes the number of successes in a series of independent Yes/No experiments all with the same ...
Array access: Find a value for array A such that A[0]=5. Bit vector arithmetic: Determine if x and y are distinct 3-bit numbers. Uninterpreted functions: Find values for x and y such that () = and () =. Most SMT solvers support only quantifier-free fragments of their logics. [citation needed]
Mathematics can be used to study Sudoku puzzles to answer questions such as "How many filled Sudoku grids are there?", "What is the minimal number of clues in a valid puzzle?" and "In what ways can Sudoku grids be symmetric?" through the use of combinatorics and group theory. The analysis of Sudoku is generally divided between analyzing the properties of unsolved puzzles (such as the minimum ...
Here are the first few cases of the binomial theorem: In general, for the expansion of (x + y)n on the right side in the n th row (numbered so that the top row is the 0th row): after combining like terms, there are n + 1 terms, and their coefficients sum to 2n. An example illustrating the last two points: with .
In the maximum metric, the distance between two points is the maximum of the absolute values of differences of their x - and y -coordinates. The last two metrics appear, for example, in routing a machine that drills a given set of holes in a printed circuit board.