I already discussed this topic in a german board. I do not think that there is a nice closed form for the probability of having x seconds to the next critical hit (related to the Pochhammer symbol, which is just a product of all the chances to have no critical hit).

However, we did monte carlo simulations to express the bonus from sharpshooter (which lets the chance to get critical hits rise) as bonus to the base chance of critical hits.

A graphical representation is here, but I think this still contains a bug we fixed later. As table, rows are attacks per second, columns are different base chance for critical hits and the entries are absolute values:

aps 5% 10% 15% 20% 25% 30% 4 0.0503 0.0374 0.0295 0.0234 0.0200 0.0170 3.8 0.0487 0.0373 0.0282 0.0211 0.0167 0.0137 3.6 0.0531 0.0399 0.0309 0.0234 0.0189 0.0152 3.4 0.0593 0.0440 0.0354 0.0294 0.0234 0.0189 3.2 0.0633 0.0456 0.0382 0.0320 0.0258 0.0212 3 0.0586 0.0471 0.0369 0.0309 0.0243 0.0206 2.8 0.061 0.045 0.0367 0.0288 0.0226 0.0176 2.6 0.0654 0.0506 0.0394 0.0334 0.0267 0.0206 2.4 0.0717 0.0569 0.0452 0.0392 0.0305 0.0282 2.2 0.0803 0.0624 0.0523 0.0423 0.0359 0.0320 2 0.0737 0.0605 0.0507 0.0419 0.0363 0.0306 1.8 0.0776 0.0607 0.0504 0.0393 0.0321 0.0266 1.6 0.0863 0.0688 0.0577 0.0462 0.0398 0.0332 1.4 0.0987 0.0847 0.0682 0.0606 0.0513 0.0449 1.2 0.1107 0.0930 0.0812 0.0727 0.0626 0.0558 1 0.1075 0.0940 0.0809 0.0684 0.0589 0.0546 0.8 0.1159 0.1013 0.0864 0.0732 0.0618 0.0542 0.6 0.1434 0.1268 0.1090 0.1000 0.0859 0.0738

Example: With 2 attacks per second and 5% base crit chance (which is the minimum), sharpshooter gives you an effective bonus of ~7,4% critchance.

However, this is just true if you shoot with single attacks and without any break. Attacks which hit multiple targets are similar to a higher attack speed, and running around from time to time increases the bonus.

Usually, you will have a critical hit much earlier. Simple example: 10 shots with ~10% chance already give a ~70% chance to have a critical hit somewhere. With 2 attacks per second, you get these 10 in the range of ~7.5-12.5%, far away from 50%.My thoughts are that on average it would take 16 and 2/3 seconds to land a critical strike since that is how long it would take for the critical hit chance to reach 50 percent assuming I went that long without landing a critical hit.

I still have my simulation code, so I can produce more values or other related data if you like.