http://www.ck12.org Chapter 4. Discrete Probability Distribution
4.4 The Binomial Probability Distribution
Learning Objectives
- Know the characteristics of the binomial random variable.
- Know the binomial probability distribution.
- Know the definitions of the mean, the variance and the standard deviation of a binomial random variable.
- Identify the type of statistical situation to which the Binomial distribution can be applied.
- Use the Binomial distribution to solve statistical problems.
Many experiments result in responses for which there is only two possible outcomes, either, a Yes or a No, Pass or
Fail, Good or Defective, Male or Female, etc. A simple example is the toss of a coin, say five times. In each toss, we
will observe either a head(H)or a tail(T). We might be interested in the probability distribution ofx, the number
of heads observed (in this case the values ofxrange from 0 to 6. Many other experiments are equivalent to the toss
of a coin, if donentimes and we are interested in the probability distributionxof times that one of the two outcomes
is observed (from 0 ton). Random variables that have this characteristic are calledbinomial random variables.
For example, let us say that we select 100 students from a large university campus and ask them whether they are in
favor of a certain issue that is going on their campus. The students are to answer with either a yes or a no. Here, we
are interested inx, the number of students who favor the issue (a Yes). If each student is randomly selected from the
total population of the university and the proportion of students who favor the issue isp, then the probability that
any randomly selected student favors the issue isp. The probability of a selected student who do not favor the issue
is 1−p. Sampling 100 students in this way is equivalent to tossing a coin 100 times.
The experiment that we have been describing is an example of abinomial experiment.It can be identified by the
following characteristics:
Characteristics of a Binomial Experiment
- The experiment consists ofnnumber of identical trials.
- There are only two possible outcomes on each trial:S(for Success) orF(for Failure).
- The probability ofSremains constant from trial to trial. We will denote it byp. We will denote the probability
ofFbyq. Thusq= 1 −p. - The trials are independent of each other.
- The binomial random variablexis the number of successes in thentrials.
Example:
In the following two examples, decide whetherxis a binomial random variable.
- Suppose a university decides to give two scholarships to two students. The pool of applicants is ten students,
six males and four females. If all the ten applicants are equally qualified and the university decides to randomly
select two. Letxbe the number of female students who receive the scholarship.