Binomial Probability Formula. In Exercises 13 and 14, answer the questions designed to help understand the rationale for the binomial probability formula.

News Source Based on data from a Harris Interactive survey, 40% of adults say that they prefer to get their news online. Four adults are randomly selected.

a. Use the multiplication rule to find the probability that the first three prefer to get their news online and the fourth prefers a different source. That is, find P(OOOD), where O denotes a preference for online news and D denotes a preference for a news source different from online.

b. Beginning with OOOD, make a complete list of the different possible arrangements of those four letters, then find the probability for each entry in the list.

c. Based on the preceding results, what is the probability of getting exactly three adults who prefer to get their news online and one adult who prefers a different news source.

A set of four adults is selected. It is given that 40% of adults prefer to get their news online.

a.

Here, O denotes an adult who prefers online news, and D denotes an adult who prefers a source other than online news.

The probability of selecting an adult who prefers online news is computed below:

$$\begin{gathered} P\left(O\right)=40\% \\ =0.40 \end{gathered}$$

The probability of selecting an adult who prefers a source different from online is computed below:

$$\begin{gathered} P\left(D\right)=1-P\left(O\right) \\ =1-0.40 \\ =0.60 \end{gathered}$$

The probability of selecting the first three adults who prefer online news and the fourth adult who prefers a source different from online is computed below:

$$\begin{gathered} P\left(OOOD\right)=P\left(O\right)P\left(O\right)P\left(O\right)P\left(D\right) \\ =\left(0.4\right)\left(0.4\right)\left(0.4\right)\left(0.6\right) \\ =0.0384 \end{gathered}$$

Thus, $P\left(OOOD\right)=0.0384$.

b.

Consider the following arrangements of a selection of four adults when exactly three of them are denoted by O and one is denoted by D:

$$\begin{gathered} OOOD \\ OODO \\ ODOO \\ DOOO \end{gathered}$$

P(OOOD) is equal to 0.0384.

Now,

$$\begin{gathered} P\left(OODO\right)=P\left(O\right)P\left(O\right)P\left(D\right)P\left(O\right) \\ =\left(0.4\right)\left(0.4\right)\left(0.6\right)\left(0.4\right) \\ =0.0384 \end{gathered}$$

Thus, P(OODO) is equal to 0.0384.

$$\begin{gathered} P\left(ODOO\right)=P\left(O\right)P\left(D\right)P\left(O\right)P\left(O\right) \\ =\left(0.4\right)\left(0.6\right)\left(0.4\right)\left(0.4\right) \\ =0.0384 \end{gathered}$$

Thus, P(ODOO) is equal to 0.0384.

$$\begin{gathered} P\left(DOOO\right)=P\left(D\right)P\left(O\right)P\left(O\right)P\left(O\right) \\ =\left(0.6\right)\left(0.4\right)\left(0.4\right)\left(0.4\right) \\ =0.0384 \end{gathered}$$

Thus, P(DOOO) is equal to 0.0384.

c.

The probability ofgetting exactly three adults who prefer to get their news online and one adult who prefers a different news source is computed below:

$$\begin{gathered} \text{Prob}=P\left(OOOD\right)+P\left(OODO\right)+P\left(ODOO\right)+P\left(DOOO\right) \\ =0.0384+0.0384+0.0384+0.0384 \\ =0.1536 \end{gathered}$$

Therefore, the probability of getting exactly three adults who prefer to get their news online and one adult who prefers a different news source is equal to 0.1536.