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Let $X_1, 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 .

My work :

My idea is to try $AM\geq GM$ inequality . 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} $. Just want to verify if my solution is correct . If it is not then can you tell where have gone wrong ?

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    $\begingroup$ Looks like you've got it! $\endgroup$ – Théophile May 15 '17 at 15:30

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