The recipe calls for four fresh eggs for each quiche. A Salt Lake County Health Department inspector paid a visit recently and pointed out that research by the Food and Drug Administration indicates that one in four eggs carries salmonella bacterium, so restaurants should never use more than three eggs when preparing quiche. The manager on duty wondered aloud if simply throwing out three eggs from each dozen and using the remaining nine in four-egg quiches would serve the same purpose.

1. Working in a group or as a class, discuss the folly of the above statement!

2. Suppose the following argument is made for three-egg quiches rather than four-egg quiches: Let x ≤ number of eggs that carry salmonella. Then p(0) = p(x = 0) = (0.75)³ = .422 for three-egg quiches and p(0) = p(x = 0) = (0.75)⁴ = .316 for four-egg quiches. What assumption must be made to justify these probability calculations? Do you think this is reasonable or not? Explain.

3. Suppose that a carton of one dozen eggs does happen to have exactly three eggs that carry salmonella and that the manager does as he proposes: selects three eggs at random and throws them out, then uses the remaining nine eggs in four-egg quiches. Let x ! number of eggs that carry salmonella among four eggs selected at random from the remaining nine.

Working with a partner, conduct a simulation to approximate the distribution of x by carrying out the following sequence of steps:

a. Take 12 identical slips of paper and write “Good” on 9 of them and “Bad” on the remaining 3. Place the slips of paper in a paper bag or some other container.

b. Mix the slips and then select three at random and remove them from the bag.

c. Mix the remaining slips and select four “eggs” from the bags.

d. Note the number of bad eggs among the four selected.

e. Replace all slips, so that the bag now contains all 12 “eggs.”

f. Repeat Steps (b)–(d) at least 10 times, each time recording the observed x value.

4. Combine the observations from your group with those from the other groups. Use the resulting data to approximate the distribution of x. Comment on the resulting distribution in the context of the risk of salmonella exposure if the manager’s proposed procedure is used.