The Journal of Consumer Research (Aug. 2011) published a study demonstrating the "last name" effect- i.e., the tendency for consumers with last names that begin with a later letter of the alphabet to purchase an item before consumers with last names that begin with earlier letters. To facilitate the analysis, the researchers assigned a number, x, to each consumer based on the first letter of the consumer's last name. For example, last names beginning with "A" were assigned x = 1, last names beginning with "B" were assigned x = 2, and last names beginning with "Z" were assigned x = 26. Do you believe the probability distribution in part a is realistic? Explain. How might you go about estimating the true probability distribution for x?

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.

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 $x=0,1,2,3$.

What dimensions do S-beams, wide-flange beams, and M-beams have in common?

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 .

Toss three fair coins, and let x equal the number of heads observed. Identify the sample points associated with this experiment, and assign a value of x to each sample point.

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.

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 standard deviation of the maximum capacity for each line. Then, give an interval for each line that will contain the maximum capacity with probability of at least .75.

What is the ignition temperature of cotton?

Process Safety Progress (Dec. 2004) published a risk analysis for a natural-gas pipeline between Bolivia and Brazil. The most likely scenario for an accident would be natural-gas leakage from a hole in the pipeline. The probability that the leak ignites immediately (causing a jet fire) is .01. If the leak does not immediately ignite, it may result in the delayed ignition of a gas cloud. Given no immediate ignition, the probability of delayed ignition (causing a flash fire) is .01. If there is no delayed ignition, the gas cloud will disperse harmlessly. Suppose a leak occurs in the natural-gas pipeline. Find the probability that either a jet fire or a flash fire will occur. Illustrate with a tree diagram.

Consider the probability distribution shown here. Use your intuition to find the mean for each distribution. How did you arrive at your choice?

$$\begin{aligned} &\begin{array}{llll} \hline x & 0 & 1 & 2 \\ p(x) & .3 & .4 & .3 \\ \hline \end{array}\\ &\begin{array}{llll} \hline y & 0 & 1 & 2 \\ p(y) & .1 & .8 & .1 \\ \hline \end{array} \end{aligned}$$

A die is tossed. Let $x$ be the number of spots observed on the upturned face of the die. Display the probability distribution of $x$ in graphical form.

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