[AlanBostick]

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I'm still amazed that people are actually are sticking with heads more than switching to tails. The chances that this coin is perfectly fair is growing smaller and smaller and it is much more likely that the coin is biased toward tails.

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Go read the problem as posed again:

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Suppose you and your friend are flipping a quarter for $10 a pop for a day. You win on heads. He wins for tails.

Somewhere in the middle, you find yourself down $160 and tails has landed 7 of the last 8 times and your friend offers you to switch to tails.

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You're tossing coins all day. Let's say you started at 9:00 AM, toss once each minute, and are continuing until 5:00 PM.


"Somewhere in the middle," i.e. around 1:00 or 2:00 PM, i.e. after three hundred tosses or so, you're stuck $160 -- that is, tails have come up 16 more times than heads has, and that the last seven tosses have been tails.

After three hundred tosses, you expect heads to have come up 150 +/- 8.7 times. Being behind by 16 tosses at midday is a two-sigma event. But note that you had just tossed tails seven times in a row. Just before that run, you had only been behind by nine tosses -- just slightly more than a one-sigma event. And a run of seven tails in a row is not terribly unlikely, with odds of just 127:1 against.

The OP doesn't specify how often the coin is tossed. If you tossed every second, rather than every minute, by 1:00 PM you would have tossed the coin 14,400 times, and would expect to see heads 7,200 +/- 60 times. Being sixteen tosses behind is well within expected results and you would also expect to have seen a number of long runs of tails and long runs of heads, and there would be no reason to even begin to suspect that the coin is biased.