# probability calculator

probability distribution

probability density

Calculates the probability mass function and lower and upper cumulative distribution functions of the hypergeometric distribution.

Probabilities

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# Poisson Distribution

Independent random events occuring in a defined time interval or a defined length, area or space volume follow Poisson distribution with parameter λ equal to the average number of events per the defined time, length, space or volume unit. The probabilty P of seeing exactly x events during the defined time interval or in the defined length, space or volume unit is given by the formula:
 P(X=x) = e-λ λx x!

λ≥0, x≥0

Calculator (fill all fields and click 'Calculate'):

Expected (average) number of events:λ =

Calculate the probability of x events:
x =
Calculate the total probability of numbers of events from the interval x1 - x2:
x1 = x2 =

P(X = x):

P(X ≤ x):

P(X ≥ x):

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# Binomial Distribution

Binomial distribution of probability describes likelihoods of all possible outcomes of n successive trials where in each trial there is the same probability p of "success" (i.e. of a defined result like e.g. a head when tossing a coin). The probabilty P of seeing exactly x successes in n successive trials is given by the formula:
 P(X=x) = n! x!(n - x)! px(1-p)n-x

n>0, 0≤x≤n, 0≤p≤1

Calculator (fill all fields and click 'Calculate'):

Number of trials:n =

Probability of success in one trial:p =

Calculate the probability of x successes in n trials:
x =
Calculate the total probability of numbers of successes from the interval x1 - x2:
x1 = x2 =

P(X = x):

P(X ≤ x):

P(X ≥ x):

Mean:

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# Hypergeometric Distribution

Hypergeometric distribution describes the probability of x successes (random draws for which the object drawn has a specified feature) in n draws, without replacement, from a finite population of size N that contains exactly M objects with that feature, wherein each draw is either a success or a failure:
P(X=x) =
 ( M x ) ( N-M n-x )
 ( N x )

N>0, M>0, M≤N, 0<n≤N, 0≤x≤M, x≤n

Calculator (fill all fields and click 'Calculate'):

Total population size:N =

Size of the sub-population with the
specified feature:M =

Sample size:n =

Calculate the probability of x successes:
x =
Calculate the total probability of values of x from the interval x1 - x2:
x1 = x2 =

P(X = x):

P(X ≤ x):

P(X ≥ x):

Mean:

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# Geometric Distribution

The geometric distribution gives the probability that the first occurrence of a success requires x independent trials, each with the success probability p. The probability P that the x-th trial is the first success is described by the formula:
 P(X=x) = p(1-p)x-1

x>0, 0<p<1

Calculator (fill all fields and click 'Calculate'):

Probability of success in one trial:p =

Calculate the probability that x trials will be required for the first success to occur:
x =
Calculate the total probability of all values x from the interval x1 - x2:
x1 = x2 =

P(X = x):

P(X ≤ x):

P(X ≥ x):

Mean:

Clear
Calculate

# Negative Binomial Distribution

Negative binomial distribution describes the probability that the x-th independent trial will be the k-th success provided each success has the same probability p:
 P(X=x) = ( x-1 k-1 ) (1-p)x-k pk
Alternatively, the distribution describes the probability of the next independent trial being the k-th success after a fixed number r of failures have occured:
 P(k, r) = ( k+r-1 k-1 ) (1-p)r pk

x>0, k>0, x≥k, r≥0, 0<p<1; x = k + r

Calculator (fill all fields and click 'Calculate'):

Probability of success in one trial:p =

Number of successes:k =

Calculate the probability that the x-th trial will be the k-th success:
x = , or alternatively, set the number of failures:r =
Calculate the total probability of all values x from the interval x1 - x2:
x1 = x2 =

P(X = x):

P(X ≤ x):

P(X ≥ x):

Mean:

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