agentcom wrote:Keep in mind that the probability of getting 70 different maps in 70 tries is pretty low. Somewhere on the order of 1 in a million tries. In other words, you're almost certain to have some repeats. You can figure this out by picking one map at random to start with. Assuming 200-ish maps, you then have a 199/200 chance of the second map being unique. Then 198/200. Etc. On down to 131/200. By multiplying all these probabilities together, you'd see that the odds are pretty low of getting 70 unique maps.
But your question is more complicated. There's the easy, but misleading question of "what are my chances of getting Cairns Metro five times in 70 games?" That's easy to solve because you can use the
binomial theorem to figure out what your chances are of getting Cairns Metro 5 times out of 70 attempts. Turns out your chances of getting Cairns Metro 5 times in 70 maps is around one in 36,000 or so. Not very common, but interestingly it's far, far more common than getting 70 unique maps.
But really the above paragraph poses the wrong question. The question should be what are the chances of getting
any map 5 times (or more) out of 70 draws. I'm heading back to my stats text book right now, and I think a good way of looking at this would be the
Poisson Distribution (though I think there are other ways). Alternatively, I believe that you could multiply the by the 200 maps to approximate a solution. Either way, I end up with a probability of about 1/150 that you end up drawing a map 5 times within 70 games.
I wouldn't call this a likely result, but I don't think that it's outside the bounds of normal variation. Someone should check my math and application of statistics though because I'm no expert in this stuff.
I would argue that Poisson doesn't work, since it assumes that if there are 200 maps, you have a 1 in 200 chance to get that map. You can't assume what you are trying to prove when you try to prove it. Binomial only assumes that there is a probability, and that it is constant. Much better test here. Then again, some non-stats people call me a fuddy-duddy.
[spoiler=A more detailed explanation]You can chi-sq from a poisson; but f*ck that! Test of binomality is the way to go.
Using all the data from this thread, p(=21): 0.086597; p(>21): 0.407587
p(=22): 0.081079; p(>22): 0.326508
p(=23): 0.072593; p(>23): 0.253915
p(=24): 0.062273; p(>24): 0.191642
p(=25): 0.05127; p(>25): 0.140372 (Out of 4115 random games, we expect 20.6 Cairns metro)
Point probability= 0.05127
p(Obs>=25): 0.1916; (<25): 0.8084
p(Obs>25): 0.1404; (<=25): 0.8596
Mid-p: 0.166; 1-p: 0.834[/spoiler]
