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Discrete Random Variables: Binomial

Summary: This module describes the characteristics of a binomial experiment and the binomial probability distribution function.

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The characteristics of a binomial experiment are:

1. There are a fixed number of trials. Think of trials as repetitions of an experiment. The letter nn denotes the number of trials. The nn trials are independent and are repeated using identical conditions. Because the nn trials are independent, the outcome of one trial does not affect the outcome of any other trial.
2. There are only 2 possible outcomes, called "success" and, "failure" for each trial. The letter pp denotes the probability of a success on one trial and qq denotes the probability of a failure on one trial. p+q=1p+q=1.
3. For each individual trial, the probability, pp, of a success and probability, qq, of a failure remain the same. For example, randomly guessing at a true - false statistics question has only two outcomes. If a success is guessing correctly, then a failure is guessing incorrectly. Suppose Joe always guesses correctly on any statistics true - false question with probability p=0.6p0.6. Then, q=0.4q0.4 .This means that for every true - false statistics question Joe answers, his probability of success ( p=0.6p0.6) and his probability of failure ( q=0.4q0.4) remain the same.

The outcomes of a binomial experiment fit a binomial probability distribution. The random variable X=X= the number of successes obtained in the nn independent trials.

The mean, μμ, and variance, σ2σ2, for the binomial probability distribution is μ=np μ np and σ2=npq. σ2 npq. The standard deviation, σσ, is then σ= npq σ npq .

Any experiment that has characteristics 2 and 3 is called a Bernoulli Trial (named after Jacob Bernoulli who, in the late 1600s, studied them extensively). A binomial experiment takes place when the number of successes are counted in one or more Bernoulli Trials.

Example 1

At ABC College, the withdrawal rate from an elementary physics course is 30% for any given term. This implies that, for any given term, 70% of the students stay in the class for the entire term. A "success" could be defined as an individual who withdrew. The random variable is XX = the number of students who withdraw from the elementary physics course per term.

Example 2

Suppose you play a game that you can only either win or lose. The probability that you win any game is 55% and the probability that you lose is 45%. If you play the game 20 times, what is the probability that you win 15 of the 20 games? Here, if you define XX = the number of wins, then XX takes on the values XX = 0, 1, 2, 3, ..., 20. The probability of a success is p=0.55 p 0.55 . The probability of a failure is q=0.45 q 0.45 . The number of trials is n=20 n 20 . The probability question can be stated mathematically as P (X=15)P (X 15).

Example 3

A fair coin is flipped 15 times. What is the probability of getting more than 10 heads? Let XX = the number of heads in 15 flips of the fair coin. XX takes on the values xx = 0, 1, 2, 3, ..., 15. Since the coin is fair, pp = 0.5 and qq = 0.5. The number of trials is nn = 15. The probability question can be stated mathematically as P( X>10) P( X 10) .

Example 4

Approximately 70% of statistics students do their homework in time for it to be collected and graded. In a statistics class of 50 students, what is the probability that at least 40 will do their homework on time?

Problem 1

This is a binomial problem because there is only a success or a __________, there are a definite number of trials, and the probability of a success is 0.70 for each trial.

failure

Problem 2

If we are interested in the number of students who do their homework, then how do we define XX?

Solution

XX = the number of statistics students who do their homework on time

Problem 3

What values does XX take on?

0, 1, 2, …, 50

Problem 4

What is a "failure", in words?

Solution

Failure is a student who does not do his or her homework on time.

The probability of a success is pp = 0.70. The number of trial is nn = 50.

Problem 5

If p+q=1 p+q 1 , then what is qq?

qq = 0.30

Problem 6

The words "at least" translate as what kind of inequality?

Solution

greater than or equal to (≥)

The probability question is P( X40) P( X 40) .

Notation for the Binomial: B = Binomial Probability Distribution Function

XX ~ B( n,p)B(n,p)

Read this as " XX is a random variable with a binomial distribution." The parameters are nn and pp. nn = number of trials pp = probability of a success on each trial

Example 5

It has been stated that about 41% of adult workers have a high school diploma but do not pursue any further education. If 20 adult workers are randomly selected, find the probability that at most 12 of them have a high school diploma but do not pursue any further education. How many adult workers do you expect to have a high school diploma but do not pursue any further education?

Let XX = the number of workers who have a high school diploma but do not pursue any further education.

XX takes on the values 0, 1, 2, ..., 20 where nn = 20 and pp = 0.41. qq = 1 - 0.41 = 0.59. XX ~ B( 20,0.41) B(20,0.41)

Find P( X12). P( X 12). P(X12)=0.9738. P(X 12) 0.9738. (calculator or computer)

Using the TI-83+ or the TI-84 calculators, the calculations are as follows. Go into 2nd DISTR. The syntax for the instructions are

To calculate (XX = value): binompdf(nn, pp, number) If "number" is left out, the result is the binomial probability table.

To calculate P(Xvalue) P(X value) : binomcdf(nn, pp, number) If "number" is left out, the result is the cumulative binomial probability table.

For this problem: After you are in 2nd DISTR, arrow down to A:binomcdf. Press ENTER. Enter 20,.41,12). The result is P(X12)=0.9738 P(X 12) 0.9738 .

Note:

If you want to find P (X=12)P (X 12) , use the pdf (0:binompdf). If you want to find P (X >12)P (X > 12), use 1 - binomcdf(20,.41,12).

The probability at most 12 workers have a high school diploma but do not pursue any further education is 0.9738

The graph of XX ~ B( 20,0.41) B(20,0.41) is:

The y-axis contains the probability of XX, where XX = the number of workers who have only a high school diploma.

The number of adult workers that you expect to have a high school diploma but not pursue any further education is the mean, μ=np= (20) (0.41)= 8.2μ=np=(20)(0.41)=8.2.

The formula for the variance is σ2=npq σ2 npq. The standard deviation is σ=npq σ npq. σ= (20) (0.41) (0.59) =2.20 σ (20) (0.41) (0.59) 2.20 .

Example 6

The following example illustrates a problem that is not binomial. It violates the condition of independence. ABC College has a student advisory committee made up of 10 staff members and 6 students. The committee wishes to choose a chairperson and a recorder. What is the probability that the chairperson and recorder are both students? All names of the committee are put into a box and two names are drawn without replacement. The first name drawn determines the chairperson and the second name the recorder. There are two trials. However, the trials are not independent because the outcome of the first trial affects the outcome of the second trial. The probability of a student on the first draw is 616616. The probability of a student on the second draw is 515515, when the first draw produces a student. The probability is 615615 when the first draw produces a staff member. The probability of drawing a student's name changes for each of the trials and, therefore, violates the condition of independence.

Glossary

Bernoulli Trials:
An experiment with the following characteristics:
• There are only 2 possible outcomes called “success” and “failure” for each trial.
• The probability pp of success are the same for any trial (so the probability q = 1-pq=1-p of failure are the same for any trial).
Binomial Distribution:
A discrete random variable (RV) which arises from the Bernoulli trials with the next additional requirements. There are a fixed number, nn, of independent trials. “Independent” means that the result of any trial (for example, trial 1) in no way affects the results of all the other trials, and all trials are conducted under the same conditions. Under these circumstances the binomial RV XX size 12{X} {} is defined as the number of successes in n trials. The notation is: XX ~ B ( n , p );B(n,p); the mean is μ=np μ np , and the variance is σ 2 = npq σ 2 =npq. The probability of exactly xx successes in nn trials is P ( X = x ) = n x p x q n x P(X=x)= n x p x q n x .

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