It'd be great if you could provide as much detail as possible. Thank you so much!!!!

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 0 1, either heads or tails, be the result. Flip the coin again, and let 0 2 be the result. If 0.1 and 0 2 are the same, return to step 1. If 0 2 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? 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 0 1, either heads or tails, be the result. Flip the coin again, and let 0 2 be the result. If 0.1 and 0 2 are the same, return to step 1. If 0 2 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