2001 Toyota starts strong most of the time

But one out of 20 times it just won't start. Here's the story: 2001 Toyota Sienna van 6cyl AT FWD, 100K miles.

I turn the key and get spark and a couple hundred rpm from the starter and it drops to zero and dies.

When the key is turned to 'on' I hear a low "brrrrp", which could be the fuel pump kicking on, but I don't know for sure. Then most of the time it cranks strong and fires up to 800 rpm and idles steady.

I noticed the problem is slightly more frequent when very low on gas. Once or twice it started but barely could hold idle, it somehow sustained a 200 rpm idle.

Problem is I can't predict when it will happen so can't replicate the problem for a mechanic. The dealer said they could try to diagnose it for $90, but if they can't reproduce the problem, they just give it back and I'm out $90.

Please weigh in if you have a theory.

thanks!