Three dice

The player throwing the three dice asked G. Galilei the question: "Should I bet on the sum of 11 or the sum of 12?" What did Galilei answer him?


Hint: write down all three triples of numbers that can be thrown and:
have a total of 11
have a total of 12

and compare probabilities.

Correct result:

p₁₁ =  0.125
p₁₂ =  0.1157
v =  11

Solution:

$$\begin{gathered} n=6\cdot 6\cdot 6=216 \\ {s}_1=1+4+6=11 \\ {s}_2=1+5+5=11 \\ {s}_3=1+6+4=11 \\ {s}_4=2+3+6=11 \\ {s}_5=2+4+5=11 \\ {s}_6=2+5+4=11 \\ {s}_7=2+6+3=11 \\ {s}_8=3+2+6=11 \\ {s}_9=3+3+5=11 \\ {s}_{10}=3+4+4=11 \\ {s}_{11}=3+5+3=11 \\ {s}_{12}=3+6+2=11 \\ {s}_{13}=4+1+6=11 \\ {s}_{14}=4+2+5=11 \\ {s}_{15}=4+3+4=11 \\ {s}_{16}=4+4+3=11 \\ {s}_{17}=4+5+2=11 \\ {s}_{18}=4+6+1=11 \\ {s}_{19}=5+1+5=11 \\ {s}_{20}=5+2+4=11 \\ {s}_{21}=5+3+3=11 \\ {s}_{22}=5+4+2=11 \\ {s}_{23}=5+5+1=11 \\ {s}_{24}=6+1+4=11 \\ {s}_{25}=6+2+3=11 \\ {s}_{26}=6+3+2=11 \\ {s}_{27}=6+4+1=11 \\ {p}_{11}={\displaystyle \frac{27}{n}}={\displaystyle \frac{27}{216}}={\displaystyle \frac{1}{8}}=0.125 \end{gathered}$$

$$\begin{gathered} s_1=1+5+6=12 \\ {s}_2=1+6+5=12 \\ {s}_3=2+4+6=12 \\ {s}_4=2+5+5=12 \\ {s}_5=2+6+4=12 \\ {s}_6=3+3+6=12 \\ {s}_7=3+4+5=12 \\ {s}_8=3+5+4=12 \\ {s}_9=3+6+3=12 \\ {s}_{10}=4+2+6=12 \\ {s}_{11}=4+3+5=12 \\ {s}_{12}=4+4+4=12 \\ {s}_{13}=4+5+3=12 \\ {s}_{14}=4+6+2=12 \\ {s}_{15}=5+1+6=12 \\ {s}_{16}=5+2+5=12 \\ {s}_{17}=5+3+4=12 \\ {s}_{18}=5+4+3=12 \\ {s}_{19}=5+5+2=12 \\ {s}_{20}=5+6+1=12 \\ {s}_{21}=6+1+5=12 \\ {s}_{22}=6+2+4=12 \\ {s}_{23}=6+3+3=12 \\ {s}_{24}=6+4+2=12 \\ {s}_{25}=6+5+1=12 \\ {p}_{12}={\displaystyle \frac{25}{n}}={\displaystyle \frac{25}{216}}=0.1157 \end{gathered}$$

$$\begin{gathered} p_{11}>p_{12} \\ v=11 \end{gathered}$$

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