The prefix

- n, the number of trials which are performed
- p, the probability of success on a single trial

In order to calculate binomial probabilties, it is necessary to know the number of ways k successes among n trials can occur. For example, the sequences of outcomes SSFSF, SSSFF, and FSSFS all entail three successes (and two failures). Typesetting requires that I use the notation C(n,k) (read as 'the number of combinations of n object taken k at a time' or 'n choose k') for the number of arrangements of k S's and (n-k) F's.

C(n,k) = n!/(k!(n-k)!)

where ! denotes factorial (e.g., 5!= 5 × 4 × 3 × 2 × 1 = 120; 3! = 3 × 2 × 1 = 6; 0! = 1 by definition). C(5,3) = 5!/(3! × 2!) = 120/(6 × 2) = 10.In order to calculate the probability of exactly k successes note first that the probability of k sucesses (and n-k failures) does not depend on the order of occurrence of the successes and failures:

P(SFSSF)=p(1-p)pp(1-p)=p^3 × (1-p)^2 = p(1-p)(1-p)pp=P(SFFSS) = etc.

The assumption of independence is necessary to get the probabilities of the sequences of outcomes by multiplication.This reduces the probability of exactly k successes to the number of arrangements of k S's and (n-k) F's times the probability of a given arrangement. The number of arrangements is C(n,k) (which is called the binomial coefficient). The probability of exactly k successes (given n trials) is

P(X=k)=C(n,k)p^k × (1-p)^(n-k)

Example Assume that 25% of fuses are defective, and the fuses in packages of six fuses are independently selected.- What is the probability that (exactly) two fuses in a package of six are defective? C(6,2)=6!/(2!4!)=720/(2 × 24) = 15; 15 × .25^2 × .75^4 = .2966.
- What is the probability that fewer than two are defective? Fewer than
two means 0 or 1. P(X=0)=C(6,0) × .25^0 × .75^6 = 1 × 1 ×
.1780 = .1780; P(X=1)=C(6,1) × .25^1 × .75^5 = 6 × .25 × .2373 = .3560. P(X=0 or 1) = .1780+.3560 = .5340.
**Competency:**If 1/5 of the jelly beans in a large bag are licorice, what is the probability that exactly 2 in a handful of 5 jelly beans are licorice? At most 2? More than 2?