»
 Exact binomial probabilities

»
 Approximation via the normal distribution

»
 Approximation via the Poisson Distribution

This unit
will calculate
and/or estimate binomial probabilities for situations of the general
"k out of n" type,
where k is the number of times a binomial outcome is observed or stipulated to occur,
p is the probability that the outcome will occur on any particular occasion,
q is the complementary probability
(1p) that the outcome will not occur on any particular occasion, and
n is the number of occasions.
For example: In 100 tosses of a coin, with
60 "heads" outcomes observed or stipulated to occur among the
100 tosses,
 n = 100
 [the number of opportunities for a head to occur]

k = 60
 [the stipulated number of heads]

p = .5
 [the probability that a head will occur on any particular toss]

q = .5
 [the probability that a head will not occur on any particular toss]

Show Description of Methods
Method 1: If n
1000, exact binomial probabilities will be calculated through repeated applications of the standard binomial formula
_{Q}
 P_{(k out of n)} =
 n! k!(nk)!
 (p^{k})(q^{nk})

In principle,
Method 1 is preferable in all cases, since it involves direct calculation of exact binomial probabilities. Its limitation is that it is not computationally feasible with very large samples. The programming on this page is capable of performing the calculation up through
n=1000.
Method 2: If np5 and nq5, binomial probabilities will be estimated by way of the binomial approximation of the normal distribution, according to the formula_{Q}
 z =
 (k—M )±.5 

where:
M =
 np [the mean of the binomial sampling distribution]

 =
 sqrt[npq] [the standard deviation of the binomial sampling distribution]

Method 3: If n≥150 and the mean (np) and variance (npq) of the binomial sampling distribution are within 10% of each other, binomial probabilities will be estimated through repeated applications of the Poisson probability function

 ^{T}P_{(k out of n)} =
 (e^{—M})(M^{k}) k!

where:
 e =
 the base of the natural logarithms; and

M =
 np [the mean of the binomial sampling distribution]

The defining characteristic of a Poisson distribution is that its mean and variance are identical. In a binomial sampling distribution, this condition is approximated as p becomes very small, providing that n is relatively large. The programming on this page permits the Poisson procedure to be performed whenever np and npq are within 10% of each other, providing that n≥150. Do keep in mind, however, that the results of the Poisson procedure are only approximations of the true binomial probabilities, valid only in the degree that the binomial mean and variance are very close.
Hide Description of Methods
To proceed, enter the values for n, k, and p into the designated cells below, and then click the «Calculate» button. (The value of q will be calculated and entered automatically). The value entered for p can be either a decimal fraction such as .25 or a common fraction such as 1/4. Whenever possible, it is better to enter the common fraction rather than a rounded decimal fraction: 1/3 rather than .3333; 1/6 rather than .1667; and so forth.
Parameters of binomial sampling distribution:
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©Richard Lowry 2001
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