Thursday, May 10, 2012

fair-dice/fair-coin paradox (my take

Game: keep tossing a dice until you quit. Last number is what the host pays you.

I believe most player would keep tossing until they get a 6, therefore,
- Average(earning) = $6, However,
- Average(value across all tosses) = 3.5

So why the fair-dice characteristic (3.5) doesn't apply to the first average? To investigate the average, we use machines -- a video-cam "recorder" to record one toss at a time.

- One recorder, the all-toss recorder, records all tosses blindly. It will probably show Average 3.5.
- One recorder, the earning recorder, doesn't record "every" toss blindly. The 1s and 2s aren't recorded. Instead, the player controls when to record. So she turns on the recorder AFTER her last toss of a game. Recorder will probably show Average 6.

Conclusion -- You can rely on the fair-dice statistical properties only if recorder is unbiased.

Q: is the recorder started before or after a toss?
If ALWAYS "before" ==> unbiased.
If sometimes "after" ==> then biased. For example, if someone records just his 4s then the recorder would include nothing but 4s -- Completely biased.
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Here's a similar paradox -- In a country where every family wants to have a boy everyone keeps having a child until they have a boy, at which point they stop. What is the proportion of boys to girls?
A: 50/50. Average ratio = 1.0, due to the unbiased all-toss recorder.

What if we only count the last child -- What's the proportion of boys to girls? Not 1.0 any more. Reason: the recorder starts AFTER the toss -- biased.

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New York (Time Square), NY, United States
http://www.linkedin.com/in/tanbin