4.4 The Binomial Distribution

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 from the course.  The random variable [latex]X[/latex] is the number of students who withdraw from the randomly selected elementary physics class.

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%.  Each game you play is independent.  If you play the game 20 times, write the function that describes the probability that you win 15 of the 20 times.

Solution:

If you define [latex]X[/latex] as the number of wins, then [latex]X[/latex] takes on the values 0, 1, 2, 3, …, 20.  The probability of a success is [latex]p=0.55[/latex].  The probability of a failure is [latex]1-p=0.45[/latex].  The number of trials is [latex]n=20[/latex].  The probability question can be stated mathematically as [latex]\displaystyle{P(x=15)}[/latex].

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

1. This is a binomial problem because there is only a success or a __________, there are a fixed number of trials, and the probability of a success is 0.70 for each trial.
2. If we are interested in the number of students who do their homework on time, then how do we define [latex]X[/latex]?
3. What values does [latex]x[/latex] take on?
4. What is a “failure,” in words?
5. What is the probability of “failuere”?
6. The words “at least” translate as what kind of inequality for the probability question [latex]\displaystyle{P(x....40)}[/latex].

Solution:

1. failure
2. [latex]X[/latex] is the number of statistics students who do their homework on time.
3. 0, 1, 2, …, 50
4. Failure is defined as a student who does not complete his or her homework on time.
5. [latex]1-p=0.30[/latex]
6. “At least” means greater than or equal to ([latex]\geq[/latex]). The probability question is [latex]\displaystyle{P(x\geq40)}[/latex].

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 ten staff members and six students.  The committee wishes to choose a chairperson and a recorder. What is the probability that the chairperson and recorder are both students?

Solution:

The names of all committee members 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 [latex]\displaystyle{\frac{6}{16}}[/latex] and the probability of a student on the second draw is [latex]\displaystyle{\frac{5}{15}}[/latex]. The probability of drawing a student’s name changes for each of the trials and, therefore, violates the condition of independence.

It has been stated that about 41% of adult workers have a high school diploma but do not pursue any further education.  Suppose 20 adult workers are randomly selected.

1. How many adult workers in the sample do you expect to have a high school diploma but do not pursue any further education?
2. What is the probability that exactly 8 of the workers in the sample have a high school diploma but do not pursue further education?
3. What is the probability that at most 12 of the workers in the sample have a high school diploma but do not purse further education?

Solution:

Let [latex]X[/latex] be the number of workers in the sample who have a high school diploma but do not pursue further education.  The number of trials is [latex]n=20[/latex] and the probability of success is [latex]p=0.41[/latex].

1. [latex]\displaystyle{\mu=n \times p=20 \times 0.41=8.2}[/latex].  On average, in any sample of 20 workers, 8.2 have a high school diploma but do not pursue further education.

2. We want to find [latex]\displaystyle{P(x=8)}[/latex].

The probability that exactly 8 of the workers in the sample have a high school diploma but do not pursue further education is [latex]17.9\%[/latex].

3. We want to find [latex]\displaystyle{P(x \leq 12)}[/latex].

The probability that at most 12 of the workers in the sample have a high school diploma but do not pursue further education is [latex]97.38\%[/latex].

In the 2013 Jerry’s Artarama art supplies catalog, there are 560 pages and 1.5% of the pages feature signature artists.  Suppose 100 pages are randomly selected from the catalog.

1. What is the probability that fewer than 3 of the pages in the sample feature signature artists?
2. What is the probability that more than 5 of the pages in the sample feature signature artists?
3. What is the probability that at least 4 of the pages in the sample feature signature artists?
4. What is the probability that between 2 and 6 of the pages in the sample feature signature artists?

Solution:

1. We want to find [latex]\displaystyle{P(x \lt 3)}[/latex].  We cannot find this probability directly in Excel because the binom.dist function can only calculate [latex]=[/latex] or [latex]\leq[/latex] probabilities.  Because [latex]x[/latex] must be an integer (it is the number of pages), [latex]x \lt 3[/latex] is the same as [latex]x \leq 2[/latex] (of course, in general, this is not true).  So [latex]\displaystyle{P(x \lt 3)=P(x \leq 2)}[/latex] and [latex]\displaystyle{P(x \leq 2)}[/latex] is a probability we can calculate with the binom.dist functon.
   

   Function	binom.dist	Answer
   Field 1	2	0.8098
   Field 2	100
   Field 3	0.015
   Field 4	true

2.  We want to find [latex]\displaystyle{P(x \gt 5)}[/latex].   We cannot find this probability directly in Excel because the binom.dist function can only calculate [latex]=[/latex] or [latex]\leq[/latex] probabilities.  The complement of [latex]\gt[/latex] is [latex]\leq[/latex], so [latex]\displaystyle{P(x \gt 5)=1-P(x \leq 5)}[/latex] and [latex]\displaystyle{P(x \leq 5)}[/latex] is a probability we can calculate with the binom.dist function.
   

   Function	1-binom.dist	Answer
   Field 1	5	0.0177
   Field 2	100
   Field 3	0.015
   Field 4	true

3. We want to find [latex]\displaystyle{P(x \geq 4)}[/latex].  We cannot find this probability directly in Excel because the binom.dist function can only calculate [latex]=[/latex] or [latex]\leq[/latex] probabilities.  The complement of [latex]\geq[/latex] is [latex]\lt[/latex], so [latex]\displaystyle{P(x \geq 4)=1-P(x  \lt 4)}[/latex].  Because [latex]x[/latex] must be an integer (it is the number of pages), [latex]x \lt 4[/latex] is the same as [latex]x \leq 3[/latex].  So [latex]\displaystyle{P(x\geq 4)=1-P(x\lt 4)=1-P(x\leq 3)}[/latex] and [latex]\displaystyle{P(x \leq 3)}[/latex] is a probability we can calculate with the binom.dist function.
   

   Function	1-binom.dist	Answer
   Field 1	3	0.0642
   Field 2	100
   Field 3	0.015
   Field 4	true

4.  We want to find [latex]\displaystyle{P(2 \leq x \leq 6)}[/latex].  We cannot find this probability directly in Excel because the binom.dist function can only calculate [latex]=[/latex] or [latex]\leq[/latex] probabilities.  But, [latex]\displaystyle{P(2 \leq x \leq 6)=P(x \leq 6)-P(x \leq 1)}[/latex].  So we can calculate [latex]\displaystyle{P(2 \leq x \leq 6)}[/latex] as the difference of two binom.dist functions.