PCR (Polymerase chain reaction) is an important technique critical for modern DNA sequencing.

 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.