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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.
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