Solar energy cells. According to the Earth Policy Institute (July 2014), $65 \%$ of the world's solar energy cells are manufactured in China. Consider a random sample of five solar energy cells and let $x$ represent the number in the sample that are manufactured in China. In the next section, we show that the probability distribution for $x$ is given by the formula: $p(x)=\frac{(5 !)(.65)^x(.35)^{5-x}}{(x !)(5-x) !}$, where $n !=(n)(n-1)(n-2) \cdots(2)(1)$ and $0 !=1$

Find the probability that at least four of the five solar energy cells in the sample are manufactured in China.

Solar energy cells. According to the Earth Policy Institute (July 2014), $65 \%$ of the world's solar energy cells are manufactured in China. Consider a random sample of five solar energy cells and let $x$ represent the number in the sample that are manufactured in China. In the next section, we show that the probability distribution for $x$ is given by the formula: $p(x)=\frac{(5 !)(.65)^x(.35)^{5-x}}{(x !)(5-x) !}$, where $n !=(n)(n-1)(n-2) \cdots(2)(1)$ and $0 !=1$

Find $p(x)$ for $x=0,1,2,3,4$, and 5 .

Is the square footage of a house discrete or continuous variable ?

Choosing portable grill displays. Consider a study of how people attempt to influence the choices of others by offering undesirable alternatives (Journal of Consumer Research, March 2003). Such a phenomenon typically occurs when family members propose a vacation spot, friends recommend a restaurant for dinner, and realtors show the buyer potential homes. In one phase of the study, the researcher had each of 124 college students select showroom displays for portable grills. Five different displays (representing five different-sized grills) were available, but only three displays would be selected. The students were instructed to select the displays to maximize purchases of Grill #2 (a smaller-sized grill). In how many possible ways can the three grill displays that include Grill #2 be selected from the five displays? List the possibilities.

Choosing portable grill displays. Consider a study of how people attempt to influence the choices of others by offering undesirable alternatives (Journal of Consumer Research, March 2003). Such a phenomenon typically occurs when family members propose a vacation spot, friends recommend a restaurant for dinner, and realtors show the buyer potential homes. In one phase of the study, the researcher had each of 124 college students select showroom displays for portable grills. Five different displays (representing five different-sized grills) were available, but only three displays would be selected. The students were instructed to select the displays to maximize purchases of Grill #2 (a smaller-sized grill).

The table shows the grill display combinations and the number of each selected by the 124 students. Use this information to assign reasonable probabilities to the different display combinations.

USDA chicken inspection. In Exercise $3.19$ (p. 146) you learned that one in every 100 slaughtered chickens passes USDA inspection with fecal contamination. Consider a random sample of three slaughtered chickens that all pass USDA inspection. Let $x$ equal the number of chickens in the sample that have fecal contamination. Find $p(x)$ for $x=0,1,2,3$.

Give an example of a continuous random variable that would be of interest to an economist.

Are you really being served red snapper? Red snapper is a rare and expensive reef fish served at upscale restaurants. Federal law prohibits restaurants from serving a cheaper, look-alike variety of fish (e.g., vermillion snapper or lane snapper) to customers who order red snapper. Researchers at the University of North Carolina used DNA analysis to examine fish specimens labeled “red snapper” that were purchased form vendors across the country ( Nature , July 15, 2004). The DNA tests revealed that 77% of the specimens were not red snapper, but the cheaper, look-alike variety of fish.
If there are five customers at a restaurant, all who have ordered red snapper, what is the probability that at least one customer is actually served red napper?

Are you really being served red snapper? Red snapper is a rare and expensive reef fish served at upscale restaurants. Federal law prohibits restaurants from serving a cheaper, look-alike variety of fish (e.g., vermillion snapper or lane snapper) to customers who order red snapper. Researchers at the University of North Carolina used DNA analysis to examine fish specimens labeled “red snapper” that were purchased form vendors across the country ( Nature , July 15, 2004). The DNA tests revealed that 77% of the specimens were not red snapper, but the cheaper, look-alike variety of fish. Assuming that the results of the DNA analysis are valid, what is the probability that you are actually served red snapper the next time you order it at a restaurant?

Give two examples of a discrete random variables that would be of interest to the manager of the clothing store.

Give an example of a discrete random variable of interest to a hospital nurse.

According to the Bureau of Justice Statistics (June 2006), 74% of all licensed drivers stopped by police are 25 years or older. Give a percentile ranking for the age of 25 years in the distribution of all ages of licensed drivers stopped by police.

A tea.in of industrial management university professors investigated the reliability of a manufacturing system that involves multiple production lines (Journal of Systems Sciences & Systems Engineering, Mar. 2013). An example of such a network is a system for producing integrated circuit (IC) cards with two production lines set up in sequence. Items (IC cards) first pass through Line 1 and then are processed by Line 2. Toe probability distribution of the maximum capacity level of each line is shown in the next column. Assume the lines operate independently.

$$\begin{matrix} \text{Line} & \text{Maximum Capacity, x} & \text{p(x)}\\ \text{1} & \text{0} & \text{.01}\\ \text{1} & \text{12} & \text{.02}\\ \text{1} & \text{24} & \text{.02}\\ \text{1} & \text{36} & \text{.95}\\ \text{2} & \text{0} & \text{.002}\\ \text{2} & \text{35} & \text{.002}\\ \text{2} & \text{70} & \text{.996}\\ \end{matrix}$$

Now consider the network of two production lines. What is the probability that a maximum capacity level of 30 items is maintained throughout the network?

Reliability of a manufacturing network. A team of industrial management university professors investigated the reliability of a manufacturing system that involves multiple production lines (Journal of Systems Sciences & Systems Engineering, March 2013). An example of such a network is a system for producing integrated circuit (IC) cards with two production lines set up in sequence. Items (IC cards) first pass through Line 1 , then are processed by Line 2. The probability distribution of the maximum capacity level $(x)$ of each line is shown below. Assume the lines operate independently. Find the mean maximum capacity for each line. Interpret the results practically.