Why do Pros use one Crit-Rune in their Rune page?

Solution 1:

It's basically just for the luck factor. With one crit rune you won't give up too much sustained damage, but you'll still have the ability to crit.

Now this possibility (although it's very unlikely to happen soon) can be really game-changing. A Random Crit can make the difference between getting or missing a kill, and a single kill can change the outcome of a Lane completely.

Also this can confuse the enemy. It's completely unpredictable whether someone will crit or not and because of the low crit chance, most people assume that he won't crit.

So basically: It's simply a "luck" while not giving up too much sustained damage.

Solution 2:

This doesn't explain entirely why pros use one crit rune, but it does explain why the one crit rune has more worth than most people think.

As far as I know, Crit chance in LoL doesn't follow the conventional way people think about % chance.

At 50% crit chance for example, you'd assume that every auto attack is like flipping a coin, and you'd have 50% chance to crit, and 50% chance to not, so if you attacked 10 times you'd expect to have 50% chance to crit every time. LoL uses something called pseudo-random distribution, it used to work the old 'coin toss' way, but that was changed pretty early on in 2011, here are the patch notes when it was changed to use the new system.

Here's a link to where Xypherous talks about it a bit:

Well, theoretically, I suppose you could "crit-farm" if the enemy champions cooperated enough to stand there so you could charge crits on them.

I hope that your opponent would be unwilling to do that for a very long period of time.

Answer: PRNG accounts for target selections.

So what does all this mean? If Tristana was attacking Ezreal, and she had 10% crit chance, for every auto attack she doesn't crit, her crit chance increases slightly until she does, and then resets. There's probably a formula somewhere to demonstrate this, but I imagine Riot uses their own constants, so i'll give a rough guide to what I mean.

1st Auto attack - 10% chance to crit
2nd Auto attack - 12% chance to crit
3rd Auto attack - 15% chance to crit
4th Auto attack - 18% chance to crit
5th Auto attack - 20% chance to crit
6th Auto attack - 24% chance to crit **Crit on Ezreal**
7th Auto attack - 10% chance to crit
8th Auto attack - 12% chance to crit
etc....

There's a nice post on the Dota forums showing more accurate math, if you're interested, I believe they use the same kind of system for crits.

As to why Pros take one crit rune, my current guess is as I said in my comment

I imagine the theory is very early game you rarely crit at all when just farming, but the crit chance is still building up. Come lvl ~4 when harass starts more frequent/jungler gank, the crit chance has built up to 3-5%, a lucky crit then could immediately win the lane.

and as Xypherous states

The system never guarantees a critical or a dodge, ever. It simply attempts to nudge the percentages in favor of it.

so obviously if you went into the game with 0% crit chance, you'll never crit. But if you take at least 1% crit chance, you're actually likely to crit more than 1% of the time.

Edit: Just to clarify

All figures I've made up are to demonstrate this point, do not take them as exact or even 'near' values, personally I'd imagine they would be much less than I stated, but you can't really tell unless you know Riot's exact method of determining them.

Edit 2:

It's worth noting that this system (whether or not Riot also make use of this, I don't know) applies both ways, as Mikey Mouse pointed out, which is also demonstrated on that Dota forum link earlier in this answer.

So if you had 10% critical hit chance, you get lucky and you keep on getting critical hit after critical hit, your chances to get another would actually drop each time, to try and balance this out. An example of this occurrence can be seen in the graph I've nicked from the DOTA source.

If you compare this to the coin toss example, by using pseudo-random distribution instead it's actually working against you, trying to make you crit less often to compensate for your super lucky streak.