Formula for how many photos you need to take to get one blink-free, or at least, a high probability of there being at least one blink-free.

If you're using a flash that has any pre-flash for metering purposes, that's the problem. The Minolta AF flashes are particularly terrible in this way as they cause "lazy eye" (half closed lids) very reliably in a lot of people due to their preflash-to-exposure latency. If that's happening, you need to figure out a good exposure and put the flashes on manual (so no preflash).

No one has blink reflexes fast enough to beat a manual flash. Not even with a big loud SLR mirror going off. Those using big studio lights can attest to this - you can point a massive camera with a Richter-3 mirror slap like an RZ-67 at someone, wired up to studio flashes with enough light to start a small fire... and they still can't blink in your shots. Every single shot comes out blink-free.