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Let $X_1$, and $X_2$ and $X_3$ be chosen independently from the set $\{0, 1, 2, 3, 4\}$, each value being equally likely. What is the probability that the arithmetic mean of $X_1, X_2$ and $X_3$ is the same as their geometric mean?

  1. $\frac{1}{5^2}\\$
  2. $\frac{1}{5^3}\\$
  3. $\frac{3!}{5^3}\\$
  4. $\frac{3}{5^3}$
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By $AM-GM$ we get that equality holds only when $X_1=X_2=X_3 $ . So There are only $5$ tuples such that $X_1=X_2=X_3$ .

 

So required probability :

$$\frac{5}{5^3}=\frac{1}{25} $$.

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