Learning Objectives Introduction Binomial Experiment Binomial Distribution Binomial Probabilities Binomial Moments Binomial Quantiles Binomial Approximation to the Hypergeometric Examples
The Binomial module targets the following cognitive tasks:
| Task Skills | Concepts | |
|---|---|---|
| Binom-1: | Understand Bernoulli trials | |
| Binom-2: | Identify a binomial experiment | Understand the conditions of a binomial experiment |
| Binom-3: | Identify a binomial random variable | Understand binomial random variables |
| Binom-4: | Compute binomial probabilities | Develop the binomial probability density function (pdf) |
| Binom-5: | Compute the theoretical mean of a binomial random variable | Derive the theoretical mean of a binomial random variable |
| Binom-6: | Compute the theoretical variance of a binomial random variable | Derive the theoretical variance of a binomial random variable |
| Binom-7: | Compute binomial cumulative probabilities | Develop the binomial cumulative distribution function (cdf) |
| Binom-8: | Compute binomial quantiles | Understand binomial quantiles |
Data often arise in the form of counts or proportions which are realizations of a discrete random variable. A common situation is to record how many times an event occurs in n repetitions of an experiment, i.e., for each repetition the event either occurs (a "success") or it does not (a "failure"). The probability of this event is modeled by the binomial distribution.
More specifically, consider the following experimental process:
An experiment satisfying these four conditions is called a binomial experiment. The outcome of this type of experiment is the number of successes, i.e., a count. The discrete random variable (denoted by X) representing the number of successes is called a binomial random variable. The possible counts, , and their associated probabilities define the binomial probability density function (pdf) or binomial distribution, denoted by B(n, p).
The addition and multiplication probability laws presented earlier can be used to compute the binomial probabilities for small , i.e., the probability of getting exactly
successes out of
trials. However, as
gets large, this becomes impractical.
By reasoning inductively, we can derive the binomial probability density function as:
As an example, consider a binomial experiment in which and
. Then,
The following Binomial Applet can be used to compute binomial probabilities and quantiles.
You can compute probabilities by selecting a menu item from the Prob menu and specifying a and/or b in the resulting dialog window. The relationship between the binomial pdf (or cdf) and the binomial parameters are displayed dynamically as the user changes n and p (by clicking on the left or right triangular buttons).
What is the probability of getting exactly 4 successes out of 10 trials when p = 0.4? This can be computed directly and verified with the above Binomial Applet.
The binomial mean, or the expected number of successes in trials, is:
The standard deviation is:
Example #1 shows how probabilities and quantiles are computed when a student guesses on a multiple-choice test.
Example #2 compares the distributions of the number of delinquents and non-delinquents who wear glasses.
Self-test