Math 207 – Binomial, Hypergeometric, and Poisson Distribution Examples

Note these probability distributions are another form of “modeling” (like we did with regression analysis). In these situations, we model experimental results by determining the possible outcomes and corresponding probabilities.

 

Example 1 (Binomial Distribution)

On a 20-question multiple-choice test there are 5 possible answers, of which only one answer is correct. Suppose a student guesses randomly and independently. (Note: This isn’t a very good strategy.)

  1. Verify this is a binomial setting. Then, for the rest of this example, define the random variable, X, to be the number of questions this student answers correctly.

 

 

  1. Determine the probability the student gets 50% of the test correct (use the binomial probability distribution).

 

 

  1. Determine the probability the student gets 50% or fewer of the test questions correct. Also determine the probability the student gets more than 5 correct answers (use Table 1 in the appendix).

 

 

  1. Determine the expected number of correct answers and the standard deviation of the number of correct answers.

 

 

Example 2 (Rule for Approximate Independence in the Binomial Setting)

Consider the following two situations:

 

Situation 1

Suppose we have a population of 20 widgets, of which 10% are defective (that is, 2 defective and 18 non-defective widgets). Two widgets are randomly drawn (without replacement). Let X be the number of defective widgets in the sample.

 

B—binary outcomes (defective or non-defective)

I—independence?

N—fixed number of draws (n = 2)

S—same probability of defective (p = 0.1)

 

Check the independence:  

Hence, the draws are not independent, so this is not a binomial setting.

 

Situation 2

Suppose we have a population of 100,000 widgets, of which 10% are defective (that is, 10,000 defective and 90,000 non-defective widgets). Two widgets are randomly drawn (without replacement). Let X be the number of defective widgets in the sample.

 

B—binary outcomes (defective or non-defective)

I—independence?

N—fixed number of draws (n = 2)

S—same probability of defective (p = 0.1)

 

Check the independence:  

Hence, the draws are approximately independent, so this is an approximate binomial setting.

 

 

Rule of Thumb for Approximate Independence in the Binomial Setting

In most practice applications of the binomial distribution the sampling is done without replacement (e.g., Gallup Polls), so technically the observations are not independent. But if the population size is at least 20 times the sample size, the observations are approximately independent and the binomial-distribution model can still be used.

Example 3 (Binomial versus Hypergeometric Distribution)

A large city has 250,000 adult residents, of which 64% support political candidate A. A random sample of 10 is chosen (without replacement). What is the probability that all 10 people in the sample support candidate A?

 

 

 

 

 

 

Now suppose the population is only 50, of which 64% support political candidate A. Now what is the probability that all 10 people in the sample support candidate A?

 

 

 

 

 

 

Example 4 (Poisson Distribution)

Suppose phone calls independently enter a switchboard on the average of 2 every 3 minutes.

 

  1. Determine the probability of exactly 5 calls in 9 minutes (use the Poisson probability distribution).

 

 

 

 

 

  1. Determine the probability of 5 or more calls in 9 minutes (use Table 2 in the appendix).

 

 

 

 

 

 

When to Apply the Different Discrete Distributions (Binomial, Hypergeometric, and Poisson)

As another “tool” in the toolbox, you can always check the BINS properties for any given problem. (Recall these are Binary outcomes, Independent outcomes, Number of outcomes fixed, and Same probability of “success” for each outcome.)

 

If all these properties are present in a given problem, you can use the binomial distribution (and Table 1 in the textbook appendix) to determine probabilities (and you can simply calculate and use the mean and standard deviation for the binomial). That is, you don’t need to “reinvent the wheel” and crank out the distribution and expected value (as you would have to for a general discrete random variable).

 

If all these properties are present in a given problem with the exception of the Independence property, then

  • If the independence property is met approximately (that is, if the population size is at least 20 times the sample size), then you can still use the binomial distribution to determine probabilities;
  • If the independence property is not met approximately (that is, the population size is less than 20 times the sample size), then you should use the hypergeometric distribution to determine probabilities (recall the hypergeometric probabilities are found by simply applying counting rules—specifically, combinations).

 

If the fixed Number of outcomes property is not met (e.g., recording the number of cars that go through a stoplight during a certain time period) and if you’re given the average number of events that occur in a certain period of time or space, then you can use the Poisson distribution (and Table 2 in the textbook appendix) to determine probabilities.