1. PCR (Polymerase chain reaction) is an important technique critical for modern DNA sequencing. The
idea is to use a sequence of rounds (biological reactions controlled by temperature changes) where
(roughly speaking) every strand of DNA is duplicated. Once in a while a round fails, and for simplicity
assume the amount of DNA stays constant in such round. Say we start with one billion copies of a
DNA strand. Assume a round fails with probability half (in practise the failure rate would be much
lower!). Under this simplistic model, what is the probability that there are exactly 8 billion copies
immediately after 9 rounds? Find a general formula for the probability that there are exactly x billion
copies after y rounds.
2. Suppose you posit a dierent model for a failed round, where in such round the amount
of DNA decreases by a factor two. Again assume equal probability for the failure or the success of a
round. What is the probability that there are exactly 2 billion copies immediately after nine rounds?
Find a general formula for the probability that there are exactly x copies after y rounds.