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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.

a. 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.

a. failure

b. If we are interested in the number of students who do their homework on time, then how do we define X ?

b. X = the number of statistics students who do their homework on time

c. What values does x take on?

c. 0, 1, 2, …, 50

d. What is a "failure," in words?

d. Failure is defined as a student who does not complete his or her homework on time.

The probability of a success is p = 0.70. The number of trials is n = 50.

e. If p + q = 1, then what is q ?

e. q = 0.30

f. The words "at least" translate as what kind of inequality for the probability question P ( x ____ 40).

f. greater than or equal to (≥)
The probability question is P ( x ≥ 40).

Try it

Sixty-five percent of people pass the state driver’s exam on the first try. A group of 50 individuals who have taken the driver’s exam is randomly selected. Give two reasons why this is a binomial problem.

This is a binomial problem because there is only a success or a failure, and there are a definite number of trials. The probability of a success stays the same for each trial.

Try it

During the 2013 regular NBA season, DeAndre Jordan of the Los Angeles Clippers had the highest field goal completion rate in the league. DeAndre scored with 61.3% of his shots. Suppose you choose a random sample of 80 shots made by DeAndre during the 2013 season. Let X = the number of shots that scored points.

  1. What is the probability distribution for X ?
  2. Using the formulas, calculate the (i) mean and (ii) standard deviation of X .
  3. Find the probability that DeAndre scored with 60 of these shots.
  4. Find the probability that DeAndre scored with more than 50 of these shots.
  1. X ~ B (80, 0.613)
    1. Mean = np = 80(0.613) = 49.04
    2. Standard Deviation = n p q = 80 ( 0.613 ) ( 0.387 ) 4.3564
  2. P ( x = 60)= 0.0036
  3. P ( x >50) = 1 – P ( x ≤ 50) = 1 – 0.6282 = 0.3718

References

“Access to electricity (% of population),” The World Bank, 2013. Available online at http://data.worldbank.org/indicator/EG.ELC.ACCS.ZS?order=wbapi_data_value_2009%20wbapi_data_value%20wbapi_data_value-first&sort=asc (accessed May 15, 2015).

“Distance Education.” Wikipedia. Available online at http://en.wikipedia.org/wiki/Distance_education (accessed May 15, 2013).

“NBA Statistics – 2013,” ESPN NBA, 2013. Available online at http://espn.go.com/nba/statistics/_/seasontype/2 (accessed May 15, 2013).

Newport, Frank. “Americans Still Enjoy Saving Rather than Spending: Few demographic differences seen in these views other than by income,” GALLUP® Economy, 2013. Available online at http://www.gallup.com/poll/162368/americans-enjoy-saving-rather-spending.aspx (accessed May 15, 2013).

Pryor, John H., Linda DeAngelo, Laura Palucki Blake, Sylvia Hurtado, Serge Tran. The American Freshman: National Norms Fall 2011 . Los Angeles: Cooperative Institutional Research Program at the Higher Education Research Institute at UCLA, 2011. Also available online at http://heri.ucla.edu/PDFs/pubs/TFS/Norms/Monographs/TheAmericanFreshman2011.pdf (accessed May 15, 2013).

“The World FactBook,” Central Intelligence Agency. Available online at https://www.cia.gov/library/publications/the-world-factbook/geos/af.html (accessed May 15, 2013).

“What are the key statistics about pancreatic cancer?” American Cancer Society, 2013. Available online at http://www.cancer.org/cancer/pancreaticcancer/detailedguide/pancreatic-cancer-key-statistics (accessed May 15, 2013).

Chapter review

A statistical experiment can be classified as a binomial experiment if the following conditions are met:

  1. There are a fixed number of trials, n .
  2. There are only two possible outcomes, called "success" and, "failure" for each trial. The letter p denotes the probability of a success on one trial and q denotes the probability of a failure on one trial.
  3. The n trials are independent and are repeated using identical conditions.

The outcomes of a binomial experiment fit a binomial probability distribution. The random variable X = the number of successes obtained in the n independent trials. The mean of X can be calculated using the formula μ = np , and the standard deviation is given by the formula σ =   n p q .

The formula for the Binomial probability density function is

P ( x ) = n ! x ! ( n - x ) ! · p x q ( n - x )

Formula review

X ~ B ( n , p ) means that the discrete random variable X has a binomial probability distribution with n trials and probability of success p .

X = the number of successes in n independent trials

n = the number of independent trials

X takes on the values x = 0, 1, 2, 3, ..., n

p = the probability of a success for any trial

q = the probability of a failure for any trial

p + q = 1

q = 1 – p

The mean of X is μ = np . The standard deviation of X is σ = n p q .

P ( x ) = n ! x ! ( n - x ) ! · p x q ( n - x )

where P(X) is the probability of X successes in n trials when the probability of a success in ANY ONE TRIAL is p.

Use the following information to answer the next eight exercises: The Higher Education Research Institute at UCLA collected data from 203,967 incoming first-time, full-time freshmen from 270 four-year colleges and universities in the U.S. 71.3% of those students replied that, yes, they believe that same-sex couples should have the right to legal marital status. Suppose that you randomly pick eight first-time, full-time freshmen from the survey. You are interested in the number that believes that same sex-couples should have the right to legal marital status.

In words, define the random variable X .

X = the number that reply “yes”

X ~ _____(_____,_____)

What values does the random variable X take on?

0, 1, 2, 3, 4, 5, 6, 7, 8

Construct the probability distribution function (PDF).

x P ( x )

On average ( μ ), how many would you expect to answer yes?

5.7

What is the standard deviation ( σ )?

What is the probability that at most five of the freshmen reply “yes”?

0.4151

What is the probability that at least two of the freshmen reply “yes”?

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Source:  OpenStax, Introductory statistics. OpenStax CNX. Aug 09, 2016 Download for free at http://legacy.cnx.org/content/col11776/1.26
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