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comment by Devac
Devac  ·  891 days ago  ·  link  ·    ·  parent  ·  post: The fake coin probability problem

Just to clarify: are you picking a coin at random before each flip or are you making this choice once and repeat toss with same coin k times?

EDIT: here http://imgur.com/rBySo2g is my diagram solution with probabilities in case you make choice once. Excuse potato quality.




mike  ·  891 days ago  ·  link  ·  

Yes, you flip the same coin 3 times. Your diagram is a good start, but what is the probability of heads on the next toss?

Devac  ·  891 days ago  ·  link  ·  

Then probability for having consecutive k Heads is in your case

0.5 * (1 + 0.5 ^ k),

so k + 1-th is just

0.5 * (1 + 0.5 ^ (k + 1))

So for k = 3 it's 0.5 * (1 + 1/8) = 9/16

For k = 4 it's 0.5 * (1 + 1/16) = 17/32

As you can see, it steadily goes down, for very high number of flips it will get very close to 0.5, but still marginally above 0.5. The limit for this formula in sense of k -> Infinity is of course 0.5, but you can't flip infinitely many times.

EDIT: I think that I have messed up, but I can't put my finger on it. If it's any excuse, I was never any good with probability. How does one differentiate consecutive case from consecutive and a next flip? This smells of Markov Chain to me, but that's just a hunch.

mike  ·  891 days ago  ·  link  ·  

Something to think about: each flip of the coin where you see heads changes the probability you've got the fake coin. If you flipped the coin 100 times and got heads every time, I think you'd be about 100% certain that the next time it will be heads.

snoodog  ·  891 days ago  ·  link  ·  

I thought of it the same way Devac thought about it but given the hint it comes down to something like this.

Edits for formatting

Turn Fake ------- Real Probability

1 1/2 ---- 1 / (.5 +1)

1 1/4 ------- 1/(.25+1 )

1 1/8 --------- 1/(.125+1 )

1 1/16 -------- 1/(.0625+1 ) -94%

1 1/2^N ---------- 1/(.5^N+1 )