6.4.1. The Binomial Probability Distribution

Example 3:

representing the number of heads as flipping a fair coin twice.

.

,

representing the number of heads as flipping a fair coin 3 times.

Therefore,

,

representing the number of heads as flipping a fair coin n times.

Then,

(n combinations)

Note: the number of combinations is equivalent to the number of ways as

drawing i balls (heads) from n balls (n flips).

Example 4:

representing the number of successes over 3 trials.

Suppose the probability of the success is while the probability of failure is .

Then,

,

representing the number of successes over n trials.

Then,

From the above example, we readily describe the binomial experiment.

Properties of Binomial Experiment

X: representing the number of successes over n independent identical trials.

The probability of a success in a trial is p while the probability of a failure is (1-p).

Binomail Probability Distribution:

Let X be the random variable representing the number of successes of a Binomial experiment.

Then, the probability distribution function for X is

.

Properties of Binomial Probability Distribution:

A random variable X has the binomial probability distribution with parameter ,

then

and

.

JavaStatSoft:

Probability density:

Statistics -> Probability -> Probability Functions -> Probability Density Function (PDF)

Probability density:

Statistics -> Probability -> Probability Functions -> Cumulative Distribution Function (CDF)

Percentile (or Quantile):

Statistics -> Probability -> Probability Functions -> Percentile Function