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I have a problem that goes like this:I have one fair coin and one double heads coin. I pick one at random and flip the same coin three times. All three times it comes up heads. What is the probability that this same coin will come up heads a fourth time?

I said that the prob on the fourth throw is 1/2. Why?

The probability for the fair coin to throw 4 heads is .5 x .5^4 = 1/32. (the first .5 because of the random choice between the two coins.)

The probability of the double heads throwing 4 heads is .5 x 1^4 = .5

Therefore, the probability of throwing a 4th heads is .5, because it is much more likely that I have the double heads coin in my hand than the fair coin.

My question is, did I make an error in figuring this? It seems too easy, but I can not find any flaws in my reasoning (because I am too close to it)

Thanks in advance.

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# Puzzling Prob Stats / Bayes problem

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