# probability that the arithmetic mean of $X_1,X_2$ and $X_3$ is the same as their geometric mean

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 ?

• Looks like you've got it! – Théophile May 15 '17 at 15:30