How are you guys getting the probability of these figures? I'm no math student, so I can't figure it out myself (I've been trying 26*25*24*23 like I learned in Grade 11, the last year I ever touched math, but that's tedious and leaves me to flounder a lot. Also, it's incorrect).
I think it's 26*26*26*26*26 [...] for a password that's all uppercase letters, then 52*52*52*52*52 [...] for upper and lowercase... add in numbers and characters makes it even more complex.
That's for the number of possibilities, yes (n^m, where n is the number of characters in the set you're choosing from and m is the amount of characters that compose the string.. same as with numbers or anything else, really), but not for the probability of getting it right within a certain number of tries.
Imagine you're trying to pick a certain card out of a deck. There are 52 possibilities. The first time you pick, you have a 1/52 chance of getting it right (and 51/52 of getting it right). The second time you pick, you've already limited a possibility, so now you're picking from a pool of the 51 left over, so you have a 1/51 chance of getting it right that time (50/51 of getting it wrong).
So, your chance of getting it wrong after seven tries is:
(51/52)*(50/51)*(49/50)*(48/49)*(47/48)*(46/47)*(45/46)
The same logic applies to cracking a password, unless you're clueless enough to attempt the same password twice (a mistake I accidentally made in my original math, since I was assuming you were trying random passwords each time.. which is because, because you're not; you're eliminating the ones you've already tried).
You could represent the above using a product series to make it look nicer, but you get the point.
If you just want to know how long it would take to have a 50% chance of getting it right, that's easier, since intuitively it's obvious (and happens to be correct) that if you try half of the possibilities, you have 50% chance of getting it right by the time you've tried that many. Of course, there's also a chance you'll get it right long before then, or long after; that 50% chance isn't the chance that you'll get it right on the exact try when you've exhausted half the possibilities, just the chance that you'll have gotten it right
by that time.