Math geekery

[blimix]

This is freaky: If you have a 97% chance of winning each hand of FreeCell, and you play one hundred hands, you are very nearly as likely to win ninety-eight of them as you are to win ninety-seven!

I did a quick simulation, running the hundred hand trial one hundred thousand times. These were the incidences of each score:

86 1
87 1
88 2
89 19
90 63
91 206
92 747
93 2095
94 4953
95 10165
96 16990
97 22710
98 22351
99 14778
100 4919

Nearly half the results are split evenly between 97 and 98.

Okay, that was really silly of me, to simulate it instead of doing the (quite simple) math. Forgive me; I was tired. The actual probabilities are:

97: 0.97⁹⁷ * 0.03³ * ₁₀₀C₃ = 0.22747

98: 0.97⁹⁸ * 0.03² * ₁₀₀C₂ = 0.22515

So the real difference is even less (by about a third) than the chart above suggests.

It makes me wonder whether my 97% win rate (over 250 games now) might just indicate a slightly lucky 96% player.

Let's see! I'll do the math for the chance of winning my 242 games (really 96.8%), at various probabilities:

95%: 0.05367
96%: 0.11350
97%: 0.13951
98%: 0.06514

(96.8%: 0.14187)

So, yeah. It's entirely possible.

If I were to refrain from playing when very tired, or when feeling careless, that score could be higher, of course. At least half of my losses have been from carelessness.

Comments

[mswae.livejournal.com]

It makes me wonder whether my 97% win rate (over 250 games now) might just indicate a slightly lucky 96% player.


So doesn't proof by mathematical induction indicate that you could just be a lucky player with a 50% win rate skill level? Or something?

[blimix.livejournal.com]

It sure does. I note that you have (aptly) changed "slightly lucky" to "lucky". But perhaps that should be "astronomically lucky". The chance of a 50% player winning 242 out of 250 games is slightly less than one in five novemdecillion. :-)

[mswae.livejournal.com]

*You* should teach at CTY! (Or anywhere.) What a great positive, educational way of saying "um, no, doofus".

;-)

[durabledora.livejournal.com]

...brain hurting.....

I suck at Freecell. It's nice to know someone can beat the stupid game.