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## Sunday, October 7, 2012

### 2-headed coin - HeardOnTheStreet & sunrise problem

See [[Heard On the Street]] Question 4.18. 10 heads in a row is very unlikely on a fair coin, so you may feel Prob(unfair) exceeds 50%.

However, it can be fairly unlikely to pick the unfair coin.

In fact, they are equally unlikely. It turns out the unlikelihood of "10 heads/fair" is numerically close to the unlikelihood of "picking an unfair". P(Fair coin picked) ~= 50/50

This brings me to the sunrise problem -- after seeing 365 * 30 (roughly 10,000) consecutive days of sun rise or 10,000 consecutive tosses showing Head, you would think P(H) is virtually 100%, i.e. you believe you have a 2-headed coin, but what if you discover there's only 1 such coin in a pool of 999,888,999,888,999,777,999 coins? Do you think you picked that one? Or do you think you had 10,000 lucky tosses? It may turn out you need more luck to pick that special coin than getting 20 heads on a fair coin. In such a case, you are more likely to have a fair coin than an unfair coin. Your next toss would be 50/50.