Binomial Derivation

Suppose that we have a situation where:

	there are n repetitions or "trials" of a random event
	each trial has two possible outcomes, "success" or "failure"
	trials are independent
	the probability of a "success", p, remains constant from trial to trial

What is the probability of obtaining r successes in n trials?

The case n=1

When n=1 there is only one trial. There is a probability p that the trial results in success (S) and probability q=1-p that the trial results in failure (F). We can show this in two ways - by a tree diagram and a table.

Successes	0	1	Total
Probability	q	p	q+p=1

The case n=2

There are four possible results from two trials: SS, SF, FS and FF. Because the trials are independent the probabilities obey the multiplication rule:

Pr(S first and S second) = Pr(S first) x Pr(S second) = Pr(S) x Pr(S) = pxp = p²

Successes	0	1	2	Total
Probability	q²	2pq	p²	q²+2pq+p²=(q+p)²=1

The case n=3

There are 8 possible results from three trials: SSS, SSF, SFS, SFF, FSS, FSF, FFS, FFF. Proceeding as before we have:

Successes	0	1	2	3	Total
Probability	q³	3pq²	3p²q	p³	q³+3pq²+3p²q+p³=(q+p)³=1

A pattern is becoming apparent in these probabilities:

The power of p is always the number of successes (remember p¹=p and p⁰=1).
The power of q is always the number of failures.
There is only one way of getting 0 successes or n successes.
The number of ways of getting r successes in n trials is the number of ways of choosing the r branches which have an S on them. There is a mathematical function which calculates this number - it is called "n choose r" and written ⁿC_r .
The individual probabilities are the terms in the series expansion of (q+p)ⁿ.
The total probability is always 1.

So, in general:

Pr( r successes in n trials) = ⁿC_r x p^r x q^(n-r)

The function ⁿC_ris probably on your calculator, but is easily calculated by hand.
Consider n=4:

⁴C₀ =⁴C₄ = 1 (by definition - see iii above)

⁴C₁ = (4/1) = 4

⁴C₂ = (4/2) x (3/1) = 6

⁴C₃ = (4/3) x (3/2) x (2/1) = 4

To continue on the recommended route click on the Graphs button at the top of the page.