1.

I don't have a point that needs validating. Before this post (the one you're reading now), I've only posted in this thread one time. Half of that post asked you to explain your math, the other half was actually making a counter point to someone you're also disagreeing with in this thread. I'm not taking sides here. I'm trying to improve the conversation.

2. Tryndamere doesn't have 10% crit. He gets 0.35% crit per point of fury. If his fury is full, he has, at a minimum, 35% crit chance. If you had built something as simple as Brawler's Gloves, this becomes 42%. With two Brawler's Gloves and full rage, Tryndamere has a 50% crit chance, and the odds of this resulting in 4 crits in a row aren't that rare.

3. Can you link us to where this information comes from? The LoLWiki (which is generally extraordinarily accurate) says nothing at all that is even remotely close to what you just described on their critical strike page. http://leagueoflegends.wikia.com/wiki/Critical_strike

4. The exact formula you used here is more or less what I was asking for, as well as some of your results. Look, you want to claim that the math says something, but you're awfully reluctant to share the math that you did with us. Wanting to say that the math says this means you're wanting to apply an unbiased scientific method to this problem. And that's great. Horay for science I say. Check other threads in Guides & Strategy that I've posted in, and you'll see I've dropped plenty of math on people myself.

But I do my math in the thread. I explain where all my numbers come from, show everyone all the formulas. And on rare occasion when I do this, someone catches a mistake I made that needs to be corrected. I correct it, recalculate, and see if the math still says basically the same thing.

You haven't done this. You've just waved your hands in the air and said "the math says," all I'm asking for is that you show us this math. Show us the formula and some of the specific results you calculated. I'll do the math myself, sure, but I want to start with error checking on math that's already been done.

I was making my last post in somewhat of a haste, so I summarized a lot. Now I have time. DrAsawa15 was the one making the suggestion and I wasn't paying attention to names, so I'm sorry about the hasty post.
I just remember being like level ~2-3, so I know I didn't have any crit from items (masteries were probably ~4%). I may have had full rage, but I can't remember. This would bring the total to around 40% max, and although it wouldn't be uncommon to get four crits in a row with that modifier, I was just getting off a little counter/harass and randomly pulled off a kill that I wasn't even aiming for before they could run away. It's the luckiest moment that I can remember in my LoL career.
Ever since Riot said that they've made crits less random, I've been trying to find out just how random. The only pattern I've noticed without a doubt is after I have IE+PD (55% crit), I can try to clear a wave and either I'll [crit then hit] each minion to death or I'll [hit then crit] each minion to death, and whichever one it is, the entire wave goes down with that exact pattern. The likelihood of this happening by chance is less than 5% per wave, and I've consistently noticed it in nearly every single game that I have noticeable crit. Even without IE, PD gives 30% alone, and [hit hit crit] or something similar can be noticed. This has lead me to my assumption about the additive nature and threshhold of the crit proc, and I haven't noticed much deviation from this assumption ever since I've made it, so I assume that the actual crit proc formula closely resembles this.
Now that I have the time, I'll drown everyone in the math. Since this will take a while, I'll just post what's in the thread now and make another post since it could take up quite some room. I believe I made the original calculations when BF Sword was 1850, so the numbers could be slightly different.

Math is done except for the parts I skipped; I can add them if someone wants to see.

The basic formula for DPS is [damage * attacks per second], so the following chart is true:
50 damage, 0.5 AS, 25 DPS
100 damage, 0.5 AS, 50 DPS
50 damage, 1.0 AS, 50 DPS
100 damage, 1.0 AS, 100 DPS
150 damage, 1.5 AS, 225 DPS

Crit needs to be added in. Using a Crit Modifier of 100%, then whenever a crit occurs, it deals double the original damage, so the following chart must be true:

100 damage, 1.0 AS, 0% crit, 100 DPS
100 damage, 1.0 AS, 50% crit, 150 DPS
100 damage, 1.0 AS, 100% crit, 200 DPS

Averaging the crit bonus and applying it to the attack damage is one way to look at the formula. My interpretation gives the modified DPS formula: [damage * {[critical strike chance * modifier] + 1} * attacks per second]

The three previous examples can be pictured as:
100 * {[0.00*1.00]+1} * 1.0 = 100
100 * {[0.50*1.00]+1} * 1.0 = 150
100 * ([1.00*1.00]+1} * 1.0 = 200

Therefore, the modified formula for DPS (assuming 100% crit mod) is [AD * {[Crit% * CritDmg]+1} * AS].

However, to calculate what item would give the most DPS, we need to figure out how much of each stat a certain amount of gold gives. By doing the division, stat per gold can be figured out.

BF Sword: 45 AD, 1650 gold, 0.02727 AD per gold
Recurve Bow: 40% AS, 1050 gold, 0.03810 AS per gold
Cloak of Agility: 18% Crit, 830 gold, 0.02169 Crit per gold

However, to figure out which purchase would be most efficient, we need to normalize the three items and determine how much our DPS would increase with each purchase. For the most accurate determination, you could use the raw stat per gold values and calculate the line equations from a massive spreadsheet, but I'm doing this by hand so assume that stat gains are purchased with 1000 gold.

0.02727 AD per gold with 1000 gold = 27.27 AD
0.03810 AS per gold with 1000 gold = 38.10 AS%
0.02169 Crit per gold with 1000 gold = 21.69 Crit%

We'll soon take our formula for DPS and plug in [stat + the amount above] three times, one for each of the three stats, and determine which purchase gives us the largest increase in DPS. This is where I made and used my program, but I'll write out the longhand until I reach 100% Crit. The thing to do before beginning the chart is convert AS% into actual [Attacks Per Second]. Most champions have a base Attacks Per Second of 0.625, so 1% AS improves Attacks Per Second by 0.00625. In this case, 38.10% AS is converted into [0.00625 * 38.10] = 0.238 attacks per second.

After modifying Crit% to fit my formula (1.00 is 100% Crit%), the stat gains are as follows:

27.27 AD
0.238 ApS
0.2169 Crit

Assuming base values of 50 AD, 0.5 AS, and 0% crit, we can begin! I'll write the first iteration in longhand, the second in shorthand, and then write the rest in condensed format.

50 AD, 0.5 AS, 0% Crit, 25 DPS before any purchase
(Parenthesis indicate the stat change)
Buying AD raises your DPS to [ (50+27.27) * {[0.00*1.00]+1} * 0.5] = 38.635
Buying AS raises your DPS to [50 * {[0.00*1.00]+1} * (0.5+0.238) ] = 36.9
Buying Crit raises your DPS to [50 * {[ (0.00+0.2169)*1.00]+1} * 0.5] = 30.4225
Therefore, 38.6 DPS is the highest, so the most efficient purchase (if your stats were 50AD / 0.5AS / 0%) is a BF Sword (or just AD in general).

Now since the best thing to get is AD, assume you made a purchase to raise your AD by an easy number to calculate; whenever I increase a stat, I use 10 AD, 0.1 AS, and 10% Crit so that the next calculations are easier. You're basically just plugging in random stat values and determining what stat is optimal to purchase for your configuration, but I'm going through in a set pattern to determine a good estimate for the ratio that you should keep your AD:AS:Crit% at. Therefore, assume that your AD has raised by 10 from the last example and continue in this pattern.

60 AD, 0.5 AS, 0% Crit, 30 DPS
AD = 43.635 DPS
AS = 44.28 DPS
Crit = 36.507 DPS
AS is the best purchase.

I'm going to continue to do the math, but the quick summary before showing it all is that Crit becomes the most efficient purchase at around 120 AD, caps at 100% at 240 AD, and AS always trails AD by about a factor of 105 (so 170 AD ≈ 1.6 AS).

(As a side note, notice thoughout the math that the same fake item purchases worth 1000 gold continue to increase your DPS more and more as your stats increase; AD is multiplicative and this is why you need to feed your AD Carry as much as you can early so that he can start scaling exponentially by midgame.)

60, 0.6, 0, 36
52.362, 50.28, 43.8084, buy AD

70, 0.6, 0, 42
58.362, 58.66, 51.1098, buy AS

70, 0.7, 0, 49
68.089, 65.66, 59.6281, buy AD

80, 0.7, 0, 56
75.089, 75.04, 68.1464, buy AD

90, 0.7, 0, 63
82.089, 84.42, 76.6647, buy AS

90, 0.8, 0, 72
93.816, 93.42, 87.6168, buy AD

100, 0.8, 0, 80
101.816, 103.8, 97.352, buy AS

100, 0.9, 0, 90
114.543, 113.8, 109.521, buy AD

110, 0.9, 0, 99
123.543, 125.18, 120.4731, buy AS

110, 1.0, 0, 110
137.27, 136.18, 133.859, buy AD

120, 1.0, 0, 120
147.27, 148.56, 146.028, buy AS

120, 1.1, 0, 132
161.997, 160.56, 160.6308, buy AD

130, 1.1, 0, 143
172.997, 173.94, 174.0167, CRIT IS MOST EFFICIENT

Notice how at 130 AD and 1.1 AS, Crit becomes the most efficient buy. This means that, in an optimal world, you should have ~130 AD and ~1.1 AS before getting even 1% Crit. Now the formula gets more complex, and I'm going to skip a lot because it's annoying to calculate the whole formula on Microsoft Calculator. Again, all this math is assuming a 100% Crit Modifier, so the mastery and IE should be considered as they give a 160% modifier together. All this means is that Crit becomes a better buy than it normally would, so you should grab Crit a bit earlier in an optimal world.

130, 1.1, 10, 157.3
190.2967, 191.334, 188.3167, buy AS

... skip ...

200, 1.8, 60, 576
654.5376, 652.16, 654.084, buy AD

... skip ...

[the following calculation assumes crit can exceed 100%, but the math still holds such that the conclusion remains true]
240, 2.1, 90, 957.6
1066.4073, 1066.128, 1066.9176, BUY CRIT

So at 240 AD and 2.1 AS, capping Crit at 100% is the most efficient purchase. Therefore, you should have ~0% Crit at 130 AD and you should have ~100% Crit at 240 AD. Shortly after, you'll cap at 2.500 Attacks Per Second (around 280-290 AD), and assuming that you have IE and the 10% Crit Mod mastery, you're going to be capped on AS, Crit%, and CritDmg, and the only purchase you can make is AD. The first time I calculated it, 120 AD and 240 AD were the benchmarks, and this time it's 130 AD and 240 AD. I think the culprit is due to the slight change in the BF Sword (50@1850 to 45@1650).

I still feel like I should note that all of these calculations assume pure auto attacks against a target dummy. Other considerations include when to get ARP, how abilities scale (most scale on AD), MS, Downtime between auto attacks (orbwalking and kiting), GA/QSS, Lifesteal, etc. All of these calculations assume an optimal world and don't consider unique passives, unique actives, or item completions: at the end of these calculations, the test character has 5 BF Swords, 7 Recurve Bows, and 5 Cloaks of Agility at the same time.

In most cases, the ADC will have 6/8 of the following items at the end of a long game: Berserker Greaves, Infinity Edge, Phantom Dancer, Trinity Force, Last Whisper, Guardian Angel, Quicksilver Sash, and The Bloodthirster. A common build order is Zerk, BT, PD, LW, GA, IE.

Alright, I'm all forums'ed out for a bit. If you have any questions, post them here and I'll get back to you.

And now the item values have changed. All this math is pointless!