Appendix to Chapter 5:Exact Binomial Probability Calculator

This page will calculate exact binomial probabilities for situations of the general "k out of N" type, through various applications of the formula
 P(k out of N) = N!k!(N-k)! (pk)(qN-k)
where:
 N = the number of opportunities for event x to occur; k = the number of times that event x occurs or is stipulated to occur; p = the probability that event x will occur on any particular occasion; and q = the probability that event x will not occur on any particular occasion.
For example: In tossing a coin 20 times, what is the probability of ending up with exactly 16 heads among the 20 tosses? In this case
 N = 20 [the number of opportunities for a head to occur] k = 16 [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]
Application of the formula using these particular values of N, k, p, and q will give the probability of getting exactly 16 heads in 20 tosses. Applying it to all values of k equal to or greater than 16 will yield the probability of getting 16 or more heads in 20 tosses, while applying it to all values of k equal to or smaller than 16 will give the probability of getting 16 or fewer heads in 20 tosses. To perform calculations of this type, enter the appropriate values for N, k, and p (the value of q will be calculated and entered automatically). Then click the "Calculate" button. To enter a new set of values for N, k, and p, click the "Reset" button. 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.
Note that this calculator will work only for cases where N is equal to or smaller than 170. For larger values of N, you can use the binomial calculator on the VassarStats site (under "Frequency Data").
 N k p q

 Details of calculation for exactly
N! =
k! =
(N-k)! =
N!/[k!(N-k)!] =
pk =
q(N-k) =
pk x q(N-k) =
p(k out of N) =

Go to Chapter 6 [Introduction to Probability Sampling Distributions]

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