Chapter 3: Q100SE (page 203)
From a production batch with 16 items, 8 items are randomly selected for quality assurance. In how many different ways can the sample be drawn? Suggest an estimate before computing the exact number.
There are 12870 different ways in which the samples can be drawn.
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The three-dice gambling problem. According toSignificance(December 2015), the 16th-century mathematician Jerome Cardan was addicted to a gambling game involving tossing three fair dice. One outcome of interest— which Cardan called a “Fratilli”—is when any subset of the three dice sums to 3. For example, the outcome {1, 1, 1} results in 3 when you sum all three dice. Another possible outcome that results in a “Fratilli” is {1, 2, 5}, since the first two dice sum to 3. Likewise, {2, 3, 6} is a “Fratilli,” since the second die is a 3. Cardan was an excellent mathematician but calculated the probability of a “Fratilli” incorrectly as 115/216 = .532.
a. Show that the denominator of Cardan’s calculation, 216, is correct. [Hint: Knowing that there are 6 possible outcomes for each die, show that the total number of possible outcomes from tossing three fair dice is 216.]
b. One way to obtain a “Fratilli” is with the outcome {1,1, 1}. How many possible ways can this outcome be obtained?
c. Another way to obtain a “Fratilli” is with an outcome that includes at least one die with a 3. First, find the number of outcomes that do not result in a 3 on any of the dice. [Hint: If none of the dice can result in a 3, then there are only 5 possible outcomes for each die.] Now subtract this result from 216 to find the number of outcomes that include at least one 3.
d. A third way to obtain a “Fratilli” is with the outcome {1, 2, 1}, where the order of the individual die outcomes does not matter. How many possible ways can this outcome be obtained?
e. A fourth way to obtain a “Fratilli” is with the outcome {1, 2, 2}, where the order of the individual die outcomes does not matter. How many possible ways can this outcome be obtained?
f. A fifth way to obtain a “Fratilli” is with the outcome {1, 2, 4}, where the order of the individual die outcomes does not matter. How many possible ways can this outcome be obtained? [Hint:There are 3 choices for the first die, 2 for the second, and only 1 for the third.]
g. A sixth way to obtain a “Fratilli” is with the outcome {1, 2, 5}, where the order of the individual die outcomes does not matter. How many possible ways can this outcome be obtained? [See Hintfor part f.]
h. A final way to obtain a “Fratilli” is with the outcome {1, 2, 6}, where the order of the individual die outcomes does not matter. How many possible ways can this outcome be obtained? [See Hintfor part f.]
i. Sum the results for parts b–h to obtain the total number of possible “Fratilli” outcomes.
j. Compute the probability of obtaining a “Fratilli” outcome. Compare your answer with Cardan’s.
Most likely coin-tossing sequence. In Parade Magazine’s (November 26, 2000) column “Ask Marilyn,” the following question was posed: “I have just tossed a [balanced] coin 10 times, and I ask you to guess which of the following three sequences was the result. One (and only one) of the sequences is genuine.”
(1) H HHHHHHHHH
(2) H H T T H T T H HH
(3) T TTTTTTTTT
Testing a watch manufacturer’s claim. A manufacturer of a new Smart Watch claims that the probability of its watch running more than 1 minute slow or 1 minute fast after 1 year of use is .05. A consumer protection agency has purchased four of the manufacturer’s watches with the intention of testing the claim.
Compute each of the following:
a.
b.
c.
d.
e.
Question: Refer to Exercise 3.35. Use the same event definitions to do the following exercises.
a. Write the event that the outcome is "On" and "High" as an intersection of two events.
b. Write the event that the outcome is "Low" or "Medium" as the complement of an event.
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