According to Significance (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.

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 Hint for 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 Hint for 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.