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Recently I was playing an online game. I was frequently beaten by someone who was performing suspiciously well. I told a friend why I didn't really like playing this game, and his response was "It happens, but I think that only 2% of these players cheat. That's not much right?"

So I got to thinking; "If only X percent of a community reliably use hacks, what is my probability on a per-game basis of encountering such a cheater on either team?"

Lets say we have a pool of 100 players, and match sizes are 10 people. Answers are rounded at the hundredths place.

In each instance, I calculate the number of ways I can choose 10 people 
    from the fair players and divide it by the total number of ways I could 
    have selected my players.

If 1 of them is a cheater, representing 1%
C(99,10) / C(100,10) = 0.9
90% Chance of a fair game, 10% chance the game contains a cheater

If 2 of them are cheaters
C(98,10) / C(100, 10) = 0.81
81% chance of a fair game, 19% chance the game contains a cheater

And if 3 of them are cheaters
C(97,10) / C(100,10) = 0.73
73% chance of a fair game, 27% chance the game contains a cheater

Is this math correct? Are percentages this small really all that's required to reliably encounter cheaters in an online game?

Note: I know "ruin" is subjective. Given the competitive nature of online games nowadays, lets say that one could consider their game experience "ruined" if they encounter a hacker on either team in > 10% of matches played.

DeepDeadpool
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1 Answers1

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Your calculations look correct to me. Your formulation is essentially "drawing without replacement" from the pool of players. A somewhat simpler way of getting a similar conclusion is to "draw with replacement", treating each player as independently having, say, a 97% chance to be playing fairly. In that model, the chance of getting a fair game would be $(.97)^{10} \approx 0.74$, yielding roughly a 26% chance of encountering a cheater, which is almost the same as the conclusion you reached in the drawing-without-replacement model (since removing a handful of non-cheaters from the population doesn't substantially change the odds of the next player being a cheater).

The drawing-with-replacement model also gives some intuition as to why the odds of a fair game are so dismal even with only a small proportion of cheaters: the probability of getting a fair game is decreasing exponentially in the team size!

Gregory J. Puleo
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