4. Suppose you want to flip a fair coin, but all you have is a biased coin that comes up heads with probability p. There is a sneaky trick for getting the fair coinflip you want anyway! The trick is: flip the coin twice, keeping track of the order. If the coin lands first heads then tails, output "heads" for the fair coinflip you want to simulate. If the coin lands first tails then heads, output "tails" for the fair coinflip you want to simulate. Otherwise, if the coin lands on the same side twice, ignore this result and start over from the beginning. (a) On a single iteration of this experiment, compute the probability of each result: output "heads", output "tails", or start over. (b) In terms of p, what is the distribution of the number of iterations required until you get an output? (Give the name and parameter(s) of the distribution.) (c) If p = 0.01 (the biased coin almost never lands heads), what is the expected number of times you'll have to flip the biased coin? (Keep in mind that we flip the coin twice on each iteration.)
4. Suppose you want to flip a fair coin, but all you have is a biased coin that comes up heads with probability p. There is a sneaky trick for getting the fair coinflip you want anyway! The trick is: flip the coin twice, keeping track of the order. If the coin lands first heads then tails, output "heads" for the fair coinflip you want to simulate. If the coin lands first tails then heads, output "tails" for the fair coinflip you want to simulate. Otherwise, if the coin lands on the same side twice, ignore this result and start over from the beginning. (a) On a single iteration of this experiment, compute the probability of each result: output "heads", output "tails", or start over. (b) In terms of p, what is the distribution of the number of iterations required until you get an output? (Give the name and parameter(s) of the distribution.) (c) If p = 0.01 (the biased coin almost never lands heads), what is the expected number of times you'll have to flip the biased coin? (Keep in mind that we flip the coin twice on each iteration.)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Transcribed Image Text:4. Suppose you want to flip a fair coin, but all you have is a biased coin that comes up heads
with probability p. There is a sneaky trick for getting the fair coinflip you want anyway!
The trick is: flip the coin twice, keeping track of the order. If the coin lands first heads
then tails, output "heads" for the fair coinflip you want to simulate. If the coin lands first
tails then heads, output "tails" for the fair coinflip you want to simulate. Otherwise, if the
coin lands on the same side twice, ignore this result and start over from the beginning.
(a) On a single iteration of this experiment, compute the probability of each result: output
"heads", output "tails", or start over.
(b) In terms of p, what is the distribution of the number of iterations required until you get
an output? (Give the name and parameter(s) of the distribution.)
(c) If p = 0.01 (the biased coin almost never lands heads), what is the expected number of
times you'll have to flip the biased coin? (Keep in mind that we flip the coin twice on
each iteration.)
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