A Bernoulli experiment is an idealized experiment where there are exactly two possible outcomes (like heads or tails) frequently denoted as success or failure. Furthermore, the result of one experiment is independent of the result of any other experiment. Got the constraints?
Suppose we do the experiment n times. The Binomial Distribution tells us the probability of k successes and n - k failures.
Here is an example. Suppose we have a fair die, and we consider success to be rolling a 6. So the probability of success on a single experiment is 1/6, and the probability of failure is 5/6.
These are usually denoted p = 1/6 and q = 1 - p = 5/6. ...........................................[edited]
With me so far? Suppose we roll three times (n = 3).
What is the probability of zero success? Because the experiments are assumed to be independent, zero successes means three failures or (5/6)(5/6)(5/6) = 125/216.
What is the probability of exactly one success? It could happen on the first roll, the second roll, or the third roll. Those are mutually exclusive so we simply add probabilities to get
(1/6)(5/6)5/6) + (5/6)(1/6)(5/6) + (5/6)5/6)(1/6) = 3 * 25/216.
What is the probability of exactly two successes? We could have success on the first and second rolls, or the first and third, or the second and third. Again, those are mutually exclusive. So
(1/6)(1/6)(5/6) + (1/6)(5/6)(1/6) + (5/6)(1/6)(1/6) = 3 * 5/216.
What is the probability of exactly three successes?
(1/6)(1/6)(1/6) = 1/ 216.
Although it is not a formal proof, there is a reasonableness check. Because you must get 0, 1, 2, or 3 successes on three rolls, the probabilities should add to 1.
125/216 + 3(25/216) + 3(5/216) + 1/216 = (125 + 75 + 15 + 1)/216 = 216/216 = 1.
Did you follow that? The general formula ASSUMING THE CONDITIONS ARE MET is
[MATH]\text {Probability of exactly k successes in n trials} = \dbinom{n}{k} * p^k * q^{n-k)}[/MATH]
