TLDR: Use this table (at bottom where it says use this table) if you want to win! Other tables have used the wrong metrics of center for misleading numbers, mine is more in-tune with what you might expect because it properly uses median. The table shows amount of gold you should expect to spend to find a specific champion. You actually only need 336 gold to get that Karthus at level 6, reroll away!

Motivation: I came across this post by u/ereklo and thought it was awesome. I’m a math degree graduate and love seeing numbers for this. However, the chart seemed off to me, especially apparent in the early levels: at level 2, do you really need to spend over 5 gold to find a particular unit? After some math, it was actually apparent that every value in the chart was inflated by a value of 2 gold, since you get a free roll at the beginning of every round! So actually at level 2, you only need to spend an average of 3.67 gold to find that Garen that you want, which is a big difference at early stages.

But the point of this post is to highlight a bigger problem regarding this table, and also every other table I’ve seen for this. Forget about the issues about not taking into account champs being held by players, etc. because these kind of realities make calculations virtually impossible. Instead, look at the table- don’t the numbers seem pretty high? To uncover this, let’s think about the following: what does average mean? In this case it turns out, it means the arithmetic mean or expected value. However, the distribution of rolls is actually geometric in nature (like flipping coins multiple times!)- basically it looks something like this:

Illustration of geometric mean

The y-axis in this plot is probability of success of say getting Gnar, and the x-axis is the number of rolls. Notice however the graph is what we call skewed right- essentially if the plot was like a hill, downhill is going to the right. In skewed graphs, the mean is actually a poor estimate of the center! This experience is definitely felt by all of you- usually, to get that one Gnar you need for your 4 wild/sorc/drag comp, you just needed to roll like 2-5 times in most scenarios- but holy cow that one game it took 30 ROLLS. If you calculated the arithmetic mean of your experiences, that one time is going to artificially bump up your expected number of rolls, and so your result might be pretty off center from what you might expect. In other words, the mean is very susceptible to outliers, whereas the median (50th percentile result) is not, because we don’t actually care about how extreme the highs and lows are, we only care about the middle number.

The numbers in the table from the original post (after adjusting by -2 gold like I mentioned earlier) would give you the mean amount of gold spent from this distribution. Therefore, we REALLY should be using the median amount of gold spent instead.

So how do we get those values? We could use math, but because I don’t trust myself, instead I programmed the problem in Python, and simulated 10,000 games to get my numbers. You can see my (uncommented rip) code here. I got both tables, My results are below- as you can see, I got a matching table from the other post (with the 2 gold correction!!) as well as the ACTUAL table that you should be using, the table from the medians. The gold values in my opinion seem much more reasonable!

USE THIS TABLE! :)

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Expected amount of gold spent for a specific champion given level and unit cost

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DON'T USE THIS, BUT VERIFIES MY RESULT WITH OTHER POST:

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https://preview.redd.it/il9spj8gfaf31.jpg?width=482&format=pjpg&auto=webp&s=a519b766079cf52edc09884a6ca8ab3547a8425a