## Part IV. Ways to Model Data

### Investigation 19: The Binomial Distribution

Binomial Distribution. A binomial distribution describes the probability, $$P$$, of a particular event, $$X$$, during a fixed number of trials, $$N$$, given the probability, $$p$$, that the event happens during a single trial. Mathematically, we express the binomial distribution as

$P(X,N)=\frac { N! }{ X!(N-X)! } \times { p }^{ X }\times { \left( 1-p \right) }^{ N-X }$

where $$!$$ is the factorial symbol. The theoretical mean, $$\mu$$, and the theoretical variance, $${ \sigma }^{ 2 }$$,for a binomial distribution are

$\mu =Np\quad \quad \qquad \quad { \sigma }^{ 2 }=Np\left( 1-p \right)$

Investigation 19. The box and whisker plot in Figure 1 includes data from the analysis of 30 samples of 1.69-oz bags of plain M&Ms. Collectively, the samples have 1699 M&Ms, of which 435 are yellow. If you pick one M&M at random from these 1699 M&Ms, what is the probability, p, that it is yellow? Suppose that this probability applies to the population of all plain M&Ms. If we draw a sample of five M&Ms from this population, what is the probability that the sample contains no yellow M&Ms? Repeat for each of 1–5 yellow M&Ms. Construct a histogram of your results and report the mean and the variance. Repeat this analysis for green M&Ms. Compare your two histograms and discuss their similarities and their differences. Using the data in Table 2, comment on the suitability of the binomial distribution for modeling the number of yellow M&Ms in samples of five M&Ms.