Copyright © Had2Know 2010-2022. All Rights Reserved.
Follow us on Facebook
Site Design by E. Emerson
Binomial Distribution Calculator
In statistics and probability, the binomial distribution's probability density function is given by the equation
PDF(x) = (nx)px(1-p)n-x,
A common situation where the binomial distribution arises is in a series of coin tosses. Suppose you flip a fair coin seven times in an attempt to get heads. In this case, n = 7 and p = 0.5. To figure the probability of tossing a head exactly four times, you evaluate
PDF(4) = (74)(0.5)4(0.5)3
To find the probability of flipping at most four heads, you compute the sum
PDF(0) + PDF(1) + PDF(2) + PDF(3) + PDF(4)
0.0078125 + 0.0546875 + 0.1640625 + 0.2734375 + 0.2734375
Another application of the binomial distribution is in rolling fair dice. For example, suppose you roll two six-sided dice to obtain a sum of 8. The probability of getting a sum of 8 with two dice is 5/36. If you roll these dice 13 times, the probability of getting an 8 exactly twice is
PDF(2) = (132 )(5/36)2(31/36)11
Binomial Mean and VarianceThe mean of the binomial distribution, μ, is given by the equation
μ = np.
The variance, σ2, is given by the equation
σ2 = np(1-p).
If you know the values of μ and σ2 but n and p are unknown, you can compute n and p with the equations
p = 1 - σ2/μ and n = μ2/(μ - σ2).
Approximation with the Normal DistributionIf n is large, the binomial distribution can be approximated by the normal distribution with a mean of np and a standard deviation of sqrt[np(1-p)]. The condition for n to be sufficiently large is subject to interpretation, but the approximation is better when n is at least 20 and p is closer to 0.5.
One rule of thumb for deciding if you can use the normal distribution is to check whether everything within 3 standard deviations from the mean is within the range of possible values. That is,
np + 3sqrt[np(1-p)] < n, and
np - 3sqrt[np(1-p)] > 0,
which simplifies to checking if n is greater than both 9p/(1-p) and 9(1-p)/p.
For instance, if you have a binomial distribution with p = 0.32 and n = 22, you can use the normal distribution to approximate the probabilities since
22 > 9(0.32)/0.68 and 22 > 9(0.68)/0.32.
© Had2Know 2010