I know it\'s a long question, but any (detailed) help would save a life! thank y

Question

I know it's a long question, but any (detailed) help would save a life! thank you!

Suppose that we want to generate a random variable X that is equally likely to be either 0 or 1, and that all we have at our disposal is a biased coin that, when flipped, lands on heads with some (unknown) probability p Consider the following procedure: Flip the coin and let 01 either heads or tails, be the result. Flip the coin again, and let 02 be the result. If 01 and 02 are the same, return to step 1. If 02 is heads, set X= 0, otherwise set X=1 Show that the random variable X generated by this procedure is equally likely to be either 0 or 1. Could we use a simpler procedure that continues to flip the coin until the last two flips are different, and then sets X=0 if the final flip is a head, and sets X= 1 if it is a tail?

Explanation / Answer

If heads = p where 1>p>0 then tails = 1-p. The outcome of 2 flips are hh,tt,ht,th. we discard the hh,tt combinations by going back to step 1. x=0 when the th combination occurs x=1 when the ht combination occurs The th combination = (1-p)*p the ht combination = p*(1-p) As these 2 are equivalent.. then the odds of a fair toss are equal. For the 2nd case we have memory of the prior flips p p p p p p p p.....and then the last 2 flips p,(1-p). P (1-p) except for p=0.5

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