The binomial probability distribution may describe the variation that occurs from one set of trials of such a binomial experiment to another.

We devote a chapter to the binomial distribution not only because it is a mathematical model for an enormous variety of real life phenomena, but also because it has important properties that recur in many other probability models.

When each number of successes X is paired with its probability of occurrence Af, the set of pairs Af, is a probability function called a binomial distribution.

The choice of P and N determines the binomial distribution uniquely, and different choices always produce different distributions ( except when Af ; ;

The set of all binomial distributions is called the family of binomial distributions, but in general discussions this expression is often shortened to `` the binomial distribution '', or even `` the binomial '' when the context is clear.

The standard deviation of a simple game like Roulette can be calculated using the binomial distribution.

In the binomial distribution, SD

The binomial distribution assumes a result of 1 unit for a win, and 0 units for a loss, rather than-1 units for a loss, which doubles the range of possible outcomes.

* Pascal distribution, a special case of the negative binomial distribution

Important and commonly encountered probability distributions include the binomial distribution, the hypergeometric distribution, and the normal distribution.

His thesis was The distribution of the binomial coefficients modulo p. He became a professor of mathematics at Whitman College in Walla Walla, Washington.

In probability theory and statistics, the negative binomial distribution is a discrete probability distribution of the number of successes in a sequence of Bernoulli trials before a specified ( non-random ) number of failures ( denoted r ) occur.

For example, if one throws a die repeatedly until the third time “ 1 ” appears, then the probability distribution of the number of non -“ 1 ” s that had appeared will be negative binomial.

The Pascal distribution ( after Blaise Pascal ) and Polya distribution ( for George Pólya ) are special cases of the negative binomial.

Then the random number of successes we have seen, X, will have the negative binomial ( or Pascal ) distribution:

Suppose we used the negative binomial distribution to model the number of days a certain machine works before it breaks down.

One has a recursive formula for binomial coefficients

More formally a k-combination of a set S is a subset of k distinct elements of S. If the set has n elements the number of k-combinations is equal to the binomial coefficient

The cane toad has many other common names, including " giant toad " and " marine toad "; the former refers to its size and the latter to the binomial name, Bufo marinus.

Thus the American bison has the binomial Bison bison ; a name of this kind would not be allowed for a plant.

As before, we say that X has a negative binomial ( or Pólya ) distribution if it has a probability mass function:

This particular remark of Khayyám and certain propositions found in his Algebra book has made some historians of mathematics believe that Khayyám had indeed a binomial theorem up to any power.

The Shamrock Quik-Mart, named after the plant of the same name ( binomial name Oxalis acetosella ), has been a main-stay for the past 22 years.

and the third power has as coefficients the triangular numbers 1, 3, 6, 10, 15, 21, ... whose term n is the binomial coefficient, so that

Although Louis Secretan's use of the name Amanita phalloides predates Link's, it has been rejected for nomenclatural purposes because Secretan's works did not use binomial nomenclature consistently ; some taxonomists have, however, disagreed with this opinion.

A binomial heap is implemented as a collection of binomial trees ( compare with a binary heap, which has a shape of a single binary tree ).

* A binomial tree of order k has a root node whose children are roots of binomial trees of orders k − 1, k − 2, ..., 2, 1, 0 ( in this order ).

Binomial trees of order 0 to 3: Each tree has a root node with subtrees of all lower ordered binomial trees, which have been highlighted.

A binomial tree of order k has 2 < sup > k </ sup > nodes, height k.

The name comes from the shape: a binomial tree of order has nodes at depth.

If the resulting merged tree has the same order as one binomial tree in one of the two heaps, then those two are merged again.

Each binomial tree has height at most log n, so this takes O ( log n ) time.

It has a better amortized running time than a binomial heap.

The ocean sunfish has various superseded binomial synonyms, and was originally classified in the pufferfish genus, as Tetraodon mola.

The binomial name is derived from Greek ; kerthios is a small tree-dwelling bird described by Aristotle and others, and brachydactyla comes from brakhus, " short " and dactulos " finger ", which refers, like the English name, to the fact that this species has shorter toes than the Common Treecreeper.

as the binomial transform of the sequence, then one has

Moments about the mean involve a binomial expansion:

When the observations are independent, this estimator has a ( scaled ) binomial distribution ( and is also the sample mean of data from a Bernoulli distribution ).

In general, the number of m-faces is equal to the binomial coefficient.

Here the quantity in parentheses is the binomial coefficient, and is equal to

An interesting consequence of the binomial theorem is obtained by setting both variables x and y equal to one.

Each binomial can then be set equal to zero, and solving for x reveals the two roots.

* Each binomial tree in a heap obeys the minimum-heap property: the key of a node is greater than or equal to the key of its parent.

and by the symmetry of the binomial coefficients, these two dimensionalities are in fact equal.

all of whose non-leading coefficients are divisible by p by properties of binomial coefficients, and whose constant coefficient equal to p, and therefore not divisible by p < sup > 2 </ sup >.

By counting the number of times 1 and 2 are both used in such a sum, it is evident that F ( n, k ) is equal to the binomial coefficient

The sum is taken over all combinations of nonnegative integer indices k < sub > 1 </ sub > through k < sub > m </ sub > such that the sum of all k < sub > i </ sub > is n. That is, for each term in the expansion, the exponents of the x < sub > i </ sub > must add up to n. Also, as with the binomial theorem, quantities of the form x < sup > 0 </ sup > that appear are taken to equal 1 ( even when x equals zero ).

More abstractly and generally, we note that the two quantities asserted to be equal count the subsets of size k and n − k, respectively, of any n-element set S. There is a simple bijection between the two families F < sub > k </ sub > and F < sub > n − k </ sub > of subsets of S: it associates every k-element subset with its complement, which contains precisely the remaining n − k elements of S. Since F < sub > k </ sub > and F < sub > n − k </ sub > have the same number of elements, the corresponding binomial coefficients must be equal.

The probability of obtaining h heads in N tosses of a coin with a probability of heads equal to r is given by the binomial distribution:

If film grains are uniformly distributed ( equal number per area ), and if each grain has an equal and independent probability of developing to a dark silver grain after absorbing photons, then the number of such dark grains in an area will be random with a binomial distribution ; in areas where the probability is low ; this distribution will be close to the classic Poisson distribution of shot noise.