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.

Step by step solution

01

Important formula

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

02

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.

03

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.

04

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 $216-125=91$.

05

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.

06

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.

07

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.

08

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.

09

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.

10

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

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

11

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

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