## Part IV. Ways to Model Data

### Investigations 20 and 21: The Poisson Distribution

Poisson Distribution. The binomial distribution is useful if we wish to model the probability of finding a fixed number of yellow M&Ms in a sample of M&Ms of fixed size, but not the probability of finding a fixed number of yellow M&Ms in a single bag.

Investigation 20. Explain why we cannot use the binomial distribution to model the distribution of yellow M&Ms in 1.69-oz bags of plain M&Ms.

To model the number of yellow M&Ms in packages of M&Ms, we use the Poisson distribution, which gives the probability of a particular event, $$X$$, given an average rate, $$\lambda$$, for that event. Mathematically, we express the Poisson distribution as

$P\left( X,\lambda \right) =\frac { { e }^{ -\lambda }{ \lambda }^{ X } }{ X! }$

The theoretical mean, $$\mu$$, and the theoretical variance, $${ \sigma }^{ 2 }$$, are both equal to $$\lambda$$.

Investigation 21. The histograms in Figure 4 include data from the analysis of 30 samples of 1.69-oz bags of plain M&Ms. Collectively, the samples have an average of 14.5 yellow M&Ms per bag. Suppose this rate applies to the population of all 1.69-oz bags of plain M&Ms. If you pick a 1.69-oz bag of plain M&Ms at random, what is the probability that it contains exactly 11 yellow M&Ms? Repeat for each of 0–29 yellow M&Ms. Construct a histogram that shows the actual distribution of bags of M&Ms for each of 0–29 yellow M&Ms, using a bin size of 1 unit, and overlay a line plot that shows the predicted distribution of bags; be sure to you use the same scale for each plot’s y-axis. Comment on your results.