Probability of being maximum value of numbers sampled from normal distribution

Question

There are n normal distributions with mean ordered from largest to smallest $m_1$, ... , $m_n$. The standard deviations are also different but not ordered. Let's define $p_i = Pr(x_i = max(x_1, x_2, ...x_n))$. Is it possible that there is a $j > 1$ such that $p_j > p_1$

Answer 1

Yes. Write $s_i$ for the standard deviations. Suppose $s_2$ is very much larger than all the other $s_i$ and very much larger than all the $m_1-m_i$. Then $p_2\approx 1/2$: if $x_2$ is greater than $m_2$ it will typically be the largest observation.

For example: $N(.1,1)$, $N(0,10000)$, $N(-0.1,1)$. Taking 10000 replicates from this I get

   1    2    3 
2750 4956 2294 

for the number of times $x_1$, $x_2$, $x_3$ is largest.