The Binomial distribution can be described as the sum of a specific number of independent trials, each of which results in either a zero or a one with constant probability. It provides a reasonable description of coin tossing experiments, among others.

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To get X heads in a sequence of N tosses, several things have to happen. If the probability of a head on a single trial is p and the probability of a tail is q=(1-p), then each sequence with X heads and N-X tails has a probability calculated by finding the product of multiplying p times itself X times (p^X), and q times itself (N-X) times (q^(N-X)), that is (p^X q^(N-X)). However, there are many sequences which match this description. By the methods of combinatorics, we can find that there are N!/(X!*(N-X)!) different combinations with X heads and N-X tails. So, the probability of X heads in N trials is

X (N-X) N! p q --------- X!* (N-X)!

**Alternative version, should be merged**

The *binomial distribution density function* provides the probability that X success will occur in N trials of a *binomial experiment*.

A *binomial experiment* has the following properties.

- 1). It consists of a sequence of N individual trials.
- 2). Two outcomes are possible for each trail, success or failure.
- 3). The probability of success on any trial, denoted p does not vary from trail to trial.
- 4). The trials are independent.

One common example of a binomial experiment is a sequence of N tosses of a coin.

Here:

- 1). This is clearly an experiment that consists of a sequence of N trials.
- 2). There are only two outcomes, Heads and Tails, Let call Heads a success and Tails a failure.
- 3). The probability of success on any one trail, denoted p, here ˝, does not vary from trial to trial.
- 4). The trails are independent – The success of the third trial, say, does not depend on the success of the 2nd trial, etc.

To find the binomial distribution function of X successes in N trials, denoted f(X), we let p denote the probability of success on any one trial. Then the probability of failure on any one trial is q=1-p. Then the probability of any particular sequence of X success (heads) out of N trials (tosses) is p^X*q^(N-X). But, there are a variety of sequences of tosses that will result in X successes out of N trials. Then, using the formula for the number of ways of obtaining X successes out of N trials, we find that the number of ways of picking X objects out of N objects that there are

- (X!/(N!*(N-X)!) ways, where
**N**!=N factorial.

- We will denote (N!/(X!*(N-X)!) by:

- (N)
- (X). (apologies…for symbolism)

Then the probability of N successes out of N trials is given by the FunctioN?:

F(X)= (N) * p^X*q^(N-X) = (N) * p^X*(1-p)^(N-X) (X) (X)

For example, consider the experiment of tossing a die with the usual 6 sides. Suppose we want to know the probability of getting three "1"s in 5 tosses. Then p=1/5. So q=4/5.

Then, using the binomial probability function for 3 successes out of 5 trials we get the probability of this occurring as:

- F(2)= (5!/(3!*2!)) * (1/5)^3 * (4/5)^2=(5*2)* (1/125)*(16/25)=(10)* (16/3023) * =160/3125=.0512

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