## Math problem: Average winning roll

Here's a question that occurred to me while I was walking home from the subway recently: What's the average winning roll in a five-man group, assuming everyone rolls? That is, if you randomly chose five numbers between one and 100, what is the expected value of the highest one?

I know a bit of statistics, but I really don't know how to begin getting at that one. However, I do know how to write a script that will calculate the answer. (Yes, these numbers are only pseudorandom, but I did some limited testing with real random numbers (from random.org) and the results were the same. Besides, I'm pretty sure Ruby's Mersenne twister pseudorandom generator is good enough for testing distributions like this.) The average winning roll out of a group of five people is ~~82.8~~ ~~83.8~~ 83.3 (tested over many, many repetitions). Now can any mathemagicians tell me why?

The graph above, in case it isn't clear, is average winning roll on the y-axis vs number of people rolling on the x-axis, tested over 100,000 trials for each group (the relatively small sample size is why the first point is not right at 50, and probably why the curve is a little wobbly).

**Edit:**I made a silly mistake and did my rolls from 0-99 instead of from 1-100. Crschmidt caught the error and provided the answer to my question: "splitting the available pool into 5 evenly distributed rolls would give you that result. (5 splits, 6 sections, 100 -(100/6) == 83.3." Thanks!

**Another good explanation by Jeffreyssai: "The average winning roll for n people is (1 - 1/n) * 100 (plus or**

Edit 2:

Edit 2:

minus a little chunkiness because you can only roll integers). It's

a royal pain explaining why with no way of typing an integral sign,

so I'll let someone else do that."

Filed under: Analysis / Opinion, Instances

## Reader Comments (Page 1 of 8)

## Holgar Dec 21st 2009 11:02AM

I'll take your word for it......... I'm sure somone whos good at math will come along to argue though.

## Daniel Dec 21st 2009 1:43PM

Talk about a boat load of fail. The OP does not understand the concept of dependency. Each roll is an independent role so average winning roll is always 50.

On a hundred sided die the median is 50. With an infinite amount of rolls the average will meet the median so that the average of all rolls is 50. If there are five players and each roll a die that is independent of the other dice then the long run average of on their independent rolls is 50.

Since the average roll over an infinite number of rolls is the same for all players it must be that this number is the average winning roll since some player must win.

The probability of a single player winning any single roll is the same: 1/5.

The only way an average winning roll could be greater than 50 is if someone is playing with a loaded die or dice.

## mtsadowski Dec 21st 2009 2:17PM

Math... Statistics... My head asplodes!

## Pandalicious Dec 21st 2009 2:35PM

The math isn't too bad. 100/number of players. Subtract that from a number and you'll get the "lowest" average winning roll. It's a bit more complicated than that, but as far as "quick and easy" is concerned, there ya go.

## Phaed Dec 21st 2009 3:05PM

This is a fun probability problem to consider, but to me it is relatively simple compared to the extent of other things that are out there.

The distribution of highest winning rolls is based on the number of people rolling and the rule that is used to select the outcome desired-- in this case, that one person just rolls higher than everyone else. I only mention this because there are different rules and choices in the selection of the 'desired result', and that influences the probabilities.

In this instance, the formula to model the average winning role rolled based on any number of people rolling on an item is (n)/(n+1) x 100, where n is the number of people rolling. By testing this formula, and comparing it to the results on your chart, you can see that they correspond pretty closely.

## tatsumasa Dec 21st 2009 4:00PM

@ daniel

you're forgetting one very important part in your mindless ranting; that would be the 'winning' part. in order for the first independent roll to win it must be greater than those four other independent rolls.

## QQinsider Dec 21st 2009 5:41PM

Daniel doesn't understand the difference between all rolls and winning rolls. Boat load of fail.

## Tanglebones Dec 21st 2009 6:19PM

@ Daniel

Assuming you're not just trolling, you're making an error in attempting to argue the gambler's fallacy in a situation where it does not apply. You're conflating "the chance that you win a roll" with "the chance that the average winning roll is 83." The chance that a given player wins an item is always one in five, However, because every win is a result of a set of five rolls, it is fairly unlikely that the median roll, 50, will win. In a given set of five rolls, it is statistically likely that two rolls will be above average. This does not in any way affect the chance of a given player to roll a certain number or to win the roll. It simply means that if a player wins the roll, then it is likely that his number fell within a range with an average of 83.

(I don't know how likely or what the range is, but i'd guess at a range of 83 plus or minus about 10 is going to be the winner somewhat more than half the time. If anyone has the mad statistical skills to figure that one out, i'd love to see it.)

## Lastcall Dec 22nd 2009 12:26AM

I would have simply explained it this way

The odds someone will roll above 90

1 man 10%

5 man 41%

10 man 65%

15 man 79%

20 man 88%

24 man 92%

25 man 93%

Basically , if you roll a "90" ....theres only an 8% chance you'll win if 24 other people are rolling too.

## mrandum Dec 22nd 2009 4:41AM

@Daniel

Bad Math Troll is bad at math.

You had to have known you were going to get served posting on a site where only the nerdiest of nerds play. And I mean that in the most loving way possible.

## Fizzlle Dec 21st 2009 11:05AM

Wow. My best friend accused me of being to into this game when I created an Excel spreadsheet to tell me which gems to assign to slots based on bonuses for best weighting. Thanks for giving me a thread I can point him to!

## Phoenix Dec 21st 2009 1:36PM

You mean you spreadsheet your character to figure out optimal gearing, right? That's what most raiding rogues do all the time. Because of the way our class is set up (GC even called us a very "mathy" class) the only way to be sure of an upgrade, or to optimize something like gemming, is to use a spreadsheet.

There's nothing wrong with it, it doesn't mean you're "way too into the game," just that you can't eyeball the gear and know right away if it's an upgrade. It's the same with several other classes, such as hunters with femaledwarf.com, and the Rawr application for everyone else.

Incidentally, this is the reason that they are simplifying gear in Cataclysm, Blizzard doesn't like that you need to plug something into a spreadsheet to see if it's actually an upgrade or not.

## elvendude Dec 21st 2009 2:55PM

Hah! That's nothing compared to my auction tracking spreadsheets. =)

Now that's true addiction!

## Mordockk Dec 21st 2009 11:06AM

You didn't need a graph to tell me if I roll a 90+ there's a much better chance, statistically, of me winning loot.

Then again, I've rolled a 99 and lost to a 100 so many times.

## Matt J Dec 21st 2009 11:32AM

That's nothing, I got my Greedy (roll 100) achievement while still losing the roll.

## jealouspirate Dec 21st 2009 11:08AM

Funny, the majority of people I meet insist that they always have horrible rolls...

## Matt J Dec 21st 2009 11:33AM

I can relate to all the horrible rollers out there, it seems I only win rolls in 5 mans like 20% of the time

## Barinthos Dec 21st 2009 11:54AM

I don't always have horrible rolls. I only have horrible rolls on items I WANT. I can roll 100 on green trash drops, but once Marrowstrike or something shows up? 1!

## Hedwinkle Dec 21st 2009 12:16PM

erm Matt J...

You do realise if you only win 20% of the time, thats a 1 in 5 chance...which is actually quite fair in a 5-man group ...

## jimbob Dec 21st 2009 12:20PM

Hedwinkle rolls to spot sarcasm

Hedwinkle rolls 0

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