Question

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.

Short answer

1. The result is true.
2. The possible outcome is 1.
3. The number of ways is 91.
4. The number of ways is 3.
5. The number of ways is 3.
6. The numbers of ways are 6.
7. The number or ways are 6.
8. The number of ways is 6.
9. The total number of outcomes is 116.
10. The probability is 0.537.

Step-by-step solution

Important formula

The formula for probability is$\mathbf{P}\mathbf{=}\frac{\mathrm{favourableoutcomes}}{\mathrm{totaloutcomes}}$

Show that the total number of possible outcomes from tossing three fair dice is 216.

If a dice is a roll the outcomes are 6. And here dice are rolled three times the outcomes are $6^3=6\times 6\times 6=216$. Hence the result is true.

Find how many possible ways this outcome can be obtained.

The result {1,1,1} comes one times in 216 outcomes.

So, the possible outcomes are 1.

Find the number of outcomes that do not result in a 3 on any of the dice.

If none of the dice can result in a 3, each die has only 5 possible outcomes.

Thus, the number of outcomes that don’t include 3 is 125.

Hence, the result is $216-125=91$.

Determine how many possible ways this outcome canbe obtained.

The outcomes are {1, 2, 1}, {1,1,2}, {2,1,1}.

Thus, the number of ways to get the outcome of {1,2,1} where the order doesn’t matter is 3.

Evaluate how many possible ways this outcome canbe obtained.

many possible ways this outcome canbe obtained.

The outcomes are {1, 2, 2}, {2, 3, 1}, {2,1,2}.

Henceforth, the number of ways to get the outcome of {1,2,2} where the order doesn’t matter is 3.

Find how many possible ways this outcome can be obtained.

The outcomes are {1, 2, 4}, {1, 4, 2}, {4, 2, 1}, {4,1, 2}, {2, 1, 4}, {2,4,1}.

Thereafter, the number of ways to get the outcome of {1, 2, 4} where the order doesn’t matter is 6.

Find the result for part g

The outcomes are {1, 2, 5}, {1, 5, 2}, {5, 2, 1}, {5,1, 2}, {2, 1, 5}, {2,5,1}.

Accordingly, the number of ways to get the outcome of {1, 2, 5} where the order doesn’t matter is 6.

Obtain the total number of possible “Fratilli” outcomes.

The outcomes are {1, 2, 6}, {1, 6, 2}, {6, 2, 1}, {6,1, 2}, {2, 1, 6}, {2,6,1}.

So, the number of ways to get the outcome of {1, 2,6} where the order doesn’t matter is 6.

Sum the results for parts b–h to obtain the total number of possible “Fratilli” outcomes.

The total no. of outcomes from part b to h is

$1+91+3+3+6+6+6=116$

Compute the probability of obtaining a “Fratilli” outcome. Compare your answer with Cardan’s.

Here the number of outcomes obtained by Fratilli is 116 and the total outcomes are 216 then

$\begin{array}{c}P=\frac{116}{216}\\ =0.537\end{array}$

Therefore,the probability is 0.537