The binomial distribution gives the discrete probability distribution of obtaining exactly successes out of Bernoulli trials (where the result of each Bernoulli trial is true with probability and false with probability ). The binomial distribution is therefore given by
(1)
 
(2)

where is a binomial coefficient. The above plot shows the distribution of successes out of trials with .
The binomial distribution is implemented in the Wolfram Language as BinomialDistribution[n, p].
The probability of obtaining more successes than the observed in a binomial distribution is
(3)

where
(4)

is the beta function, and is the incomplete beta function.
The characteristic function for the binomial distribution is
(5)

(Papoulis 1984, p. 154). The momentgenerating function for the distribution is
(6)
 
(7)
 
(8)
 
(9)
 
(10)
 
(11)

The mean is
(12)
 
(13)
 
(14)

The moments about 0 are
(15)
 
(16)
 
(17)
 
(18)

so the moments about the mean are
(19)
 
(20)
 
(21)

The skewness and kurtosis excess are
(22)
 
(23)
 
(24)
 
(25)

The first cumulant is
(26)

and subsequent cumulants are given by the recurrence relation
(27)

The mean deviation is given by
(28)

For the special case , this is equal to
(29)
 
(30)

where is a double factorial. For , 2, ..., the first few values are therefore 1/2, 1/2, 3/4, 3/4, 15/16, 15/16, ... (OEIS A086116 and A086117). The general case is given by
(31)

Steinhaus (1999, pp. 2528) considers the expected number of squares containing a given number of grains on board of size after random distribution of of grains,
(32)

Taking gives the results summarized in the following table.
0  23.3591 
1  23.7299 
2  11.8650 
3  3.89221 
4  0.942162 
5  0.179459 
6  0.0280109 
7  0.0036840 
8  
9  
10 
An approximation to the binomial distribution for large can be obtained by expanding about the value where is a maximum, i.e., where . Since the logarithm function is monotonic, we can instead choose to expand the logarithm. Let , then
(33)

where
(34)

But we are expanding about the maximum, so, by definition,
(35)

This also means that is negative, so we can write . Now, taking the logarithm of (◇) gives
(36)

For large and we can use Stirling's approximation
(37)

so
(38)
 
(39)
 
(40)
 
(41)
 
(42)

and
(43)

To find , set this expression to 0 and solve for ,
(44)

(45)

(46)

(47)

since . We can now find the terms in the expansion
(48)
 
(49)
 
(50)
 
(51)
 
(52)
 
(53)
 
(54)
 
(55)
 
(56)
 
(57)
 
(58)
 
(59)

Now, treating the distribution as continuous,
(60)

Since each term is of order smaller than the previous, we can ignore terms higher than , so
(61)

The probability must be normalized, so
(62)

and
(63)
 
(64)

Defining ,
(65)

which is a normal distribution. The binomial distribution is therefore approximated by a normal distribution for any fixed (even if is small) as is taken to infinity.
If and in such a way that , then the binomial distribution converges to the Poisson distribution with mean .
Let and be independent binomial random variables characterized by parameters and . The conditional probability of given that is
(66)

Note that this is a hypergeometric distribution.