I've played, 1000 games. I've won 51% of the time. I gain 17 elo per win. I lose 16 elo per loss. Im plat 5, 0 LP. How many games will i have to play to reach diamond v based on the 51% win rate? If you could show me how you solved it that would be cool too.
Just off of sheer logic, let's take your average elo win and divide 100 by that. You'd need to win 5.8 ~ 6 straight games to get to promos. (Perfect world scenario)
Now, let's disregard your 1000 games, as currently it's irrelevant as you've just given us your average elo increase. You're at 0 LP, 51% win rate. Let's say you're the average person, and play about 15 games a week, that's only 3 games/day, 5 days out of a 7 day week. Out of 15 games with the 51% win rate, you're winning about 7.7 games a week ~ 8 games to keep it easy, and you're losing the remaining 7. To get out of Plat 5 you'll need to have won 9 games straight, that's 5 regular games, and 3 series games. If you take the average win/loss that I gave you above, the 8 games, multiply that by 17 for the elo, you get 136 elo. Then, take the remaining games, 7 (games you lose) and multiply that by 16, you get 112. Subtract that won elo by the loss elo, as I'm unsure if you're 51% win rate means you have a long streak, or you're winning a couple then losing, and you get 24 elo. Now, divide 100 by 24 and you get 4.1111 ~ 4. That's 4 weeks straight, at a 51% win ratio, with the win loss I've listed above, for you to get out of platinum 5.
Maybe you play more, and you win more, then your required weeks decrease. Let's say you win 4, then lose 4, then win 3 and lose 2, then win 2, well, you're at 34 elo, and it would require a little less, maybe the same amount. Let's say you don't play your series games the same week as you get 100 elo, then it may take longer than 4 weeks. Regardless, unless you have a pretty consistent win rate, such as winning at least 10 games out of those 15, then it'll take you a few weeks+ to get out of one division.
This is a rough estimation. There are more factors to include, such as increased LP winnings from 17 to 20, etc, and more lost LP from 16 to 20, but this is more of a blueprint to go based off of.
This isn't actually as simple to answer as you'd think, because LP gains and losses fluctuate based on your mmr in relation to your division, and mmr gains/losses are very difficult to actually quantify now that Riot doesn't make the data publicly available. Sites like op.gg can give you approximations, but they're only that.
Also, a 51% win rate against people in Platinum V could translate into a <50% win rate against people in higher divisions.
So, ultimately, the answer to your problem wouldn't actually pertain much to the real world.
The best way is to set up a spreadsheet using if statements, to account for promos and 0LP buffers. I don't have the time or concentration to work it out fully though.
1000 instances 510 x 17 = +8670 490 x 16 = -7840
+8670 - 7840 = +830 across 1000 instances = +0.83 per instance Difference of +0.83 per 1 instance. 1 instance = 1 game Cost = 100
100/0.83 = (1/.83) x 100 = 1.205 x 100 = 120.5 games using only averages, without conditionals/breakpoints/luck/etc
Assuming you need 100 LP. Takes about half a minute to calculate.
