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September 2016

Date: 2012-01-02 07:18
Security: Public
Tags:big posts, math, steam
Subject: Musings on a math problem before sleep

I had problems getting to sleep today, because I was thinking about an interesting mathematical problem. My brain isn't going to let me sleep unless I post about it here.

For the past two weeks, Steam (the game distribution platform run by Valve) has been having a sale, as it has been doing every year at this time. This time around, there are also giveaways to encourage people to play the games it has on sale. These take the form of extra achievements for certain games. When you manage to get one of these achievements, you'll get something for your trouble. Prizes come from what Valve calls "The Great Gift Pile", and it includes games and coupons.

However, there's only a 25% chance that you'll actually get something from the GGP (as I'll call it from now on). 75% of the time, you'll receive coal instead. Coal isn't completely useless, though; if you collect 7 coal, you can 'craft' them into a random item from the GGP. In addition, each piece of coal that remains in your inventory at the end of the promotion (which is tomorrow) will count as one entry into the "Epic Holiday Giveaway", which is a big raffle which takes place at the end. The prizes for this raffle can be seen at the bottom of . Since the page is no longer online, the prizes were as follows:

  • Grand Prize (1 winner): Every single game on Steam
  • First Prize (50 winners): Top 10 items on wishlist
  • Second Prize (100 winners): Top 5 items on wishlist
  • Third Prize (1,000 winners): Valve Complete Pack
The Steam forums have had an interesting discussion going on about this last raffle. One person (whose name was "X01" on the forums) casually mentioned how having 100 coal would hardly increase your chances to win, and that you wouldn't be 100 times more likely to win. Another person replied to say that 100/<number of entries> is 100 times bigger than 1/<number of entries>, so yes, you would be 100 times more likely to win. From there, there was a big argument between X01 insisting they were correct, and most of the other people on the forum saying that, c'mon, this is basic math. It eventually ended up with X01 deleting their comments in anger and disappearing from the thread.

(If you're interested, this is the thread in question. The fun begins at post #9, but remember while reading that X01 deleted their posts, so the quotes are all there is to go on.)

At first, the answer seems really obvious. After all, with 100 coal you're 100 times more likely to win than someone who only has 1 coal... right?

The answer to the question above as stated is that yes, if you have 100 coal then you are 100 times more likely to win than someone who has 1 coal. Thus, the people arguing against X01 are right.

However, you are not 100 times more likely to win than if you only had 1 coal. Thus, X01 is also right. (There is a caveat to this, but I'll cover it later in this post.)

The thing is, the problem was never clearly stated, which is why the discussion ballooned. In reality, both groups were absolutely correct because they each viewed the problem differently:
  • X01 viewed the problem as "If you have 100 coal, are you 100 times more likely to win than if you only had 1 coal?".
  • Most other people viewed it as "If you have 100 coal, are you 100 times more likely to win than someone else who only has 1 coal?"
The reason the distinction is important is that nothing you do can affect the amount of coal anybody else gets. If someone else has 100 coal at the end of the promotion, then that can be held as always true in any calculations you make. In contrast, the amount of coal *you* get is dependent on your actions, and is thus variable. Both figures, however, contribute to the total number of entries into the raffle, which is determined by how much coal has been given out across the entire userbase.

Now, let's assume that it's the night before the raffle, and there have been 10,000,000 pieces of coal given out. Of those, one person has 100 coal and everybody else has 1 coal. From there, it's easy to conclude that if nothing changes in the following night, then the person with 100 coal is 100 times more likely to win than one of the people with 1 coal, because 100/10,000,000 is 100 times greater than 1/10,000,000.

Now, though, let's say that one of the people who only had 1 coal manages to somehow get 99 coal overnight just before the raffle ends. Thus, the total amount of coal is now 1,000,099. Is this person 100 times more likely to win than if they only had 1 coal? No, because now you're comparing 100/10,000,099 to 1/10,000,000. That's not precisely 100 times bigger; it's actually only approximately 99.99901 times bigger.

So, to conclude - both groups are absolutely right, but in reality the difference is so tiny that you may as well just call it 100 times and be done with it.

Now that I've typed all that out, I'm going to see if my brain will let me sleep now. Good night. :)

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