Question

Choosing portable grill displays. Refer to the Journal of Consumer Research (Mar. 2003) marketing study of influencing consumer choices by offering undesirable alternatives, Exercise 3.109 (p. 204). Recall that each of 124 college students selected showroom displays for portable grills. Five different displays (representing five different-sized grills) were available. Still, the students were instructed to select only three displays to maximize purchases of Grill #2 (a smaller-sized grill). The table shows the grill display combinations and the number of times each was selected by the 124 students. Suppose one of the 124 students is selected at random. Let x represent the sum of the grill numbers selected by this student. (This value indicates the size of the grills selected.)

a. Find the probability distribution for x.

b. What is the probability that x exceeds 10?

Short answer

a.

b. 0.89

Step-by-step solution

(a) Definition of probability distribution in this context

The probability distribution in this context refers to the possibility of a particular grill combination getting selected out of all the grill combinations. The associated probabilities for a particular combination to get selected are given in the chart below.

Calculation of the probability distribution

The calculation of the associated probabilities is shown below:

(b) Formula for computing P(x>10)

The formula for calculating $\mathbf{\text{P}}\mathbf{\left(}\mathbf{\text{x>10}}\mathbf{\right)}$is shown below:

$\mathbf{P}\mathbf{(}\mathbf{x}\mathbf{>10}\mathbf{)}\mathbf{=}\mathbf{P}\mathbf{(}\mathbf{x}\mathbf{=35}\mathbf{)}\mathbf{+}\mathbf{P}\mathbf{(}\mathbf{x}\mathbf{=42}\mathbf{)}\mathbf{+}\mathbf{P}\mathbf{(}\mathbf{x}\mathbf{=34}\mathbf{)}$

The $\mathbf{P}\mathbf{(}\mathbf{x}\mathbf{>10}\mathbf{)}$indicates the probability of selecting a grill display combination by more than 10 students.

Computing the P(x>10)

The calculation of the probability of selecting x more than 10 students is calculated below:

$$\begin{gathered} \mathbf{\text{P}}\mathbf{\left(}\mathbf{\text{x>10}}\mathbf{\right)}\mathbf{\text{=}}\frac{\mathbf{\text{35}}}{\mathbf{\text{124}}}\mathbf{\text{+}}\frac{\mathbf{\text{42}}}{\mathbf{\text{124}}}\mathbf{\text{+}}\frac{\mathbf{\text{34}}}{\mathbf{\text{124}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=0.28+0.34+0.27}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\text{=0.89}} \end{gathered}$$

It $\mathbf{P}\mathbf{(}\mathbf{x}\mathbf{>10}\mathbf{)}$ is found to be 0.89.