a. In a PCR you start out with N₀ molecules of template. Recognize that although PCR is an exponential amplification process, the yield of the amplification reaction is typically not 100% and we can assign a probability, p to each step of the cycle. The first few cycles when template molecules are low in number, is inefficient and later when there are sufficient copies of the template, the probability approaches 1. Starting with N₀ molecules of template, calculate total number of expected molecules (N_f) at the end of A cycles with each having a probability of p_A.

b. At the end of A cycles, the probability changes to p_B for the next B cycles. Calculate the total number of expected molecules (N_f) at the end of (a+b) cycles of PCR.

c. Look at the expression relating Nf and N0. Regardless of probability, what is the minimum number of cycles (A+B) required to achieve a billion fold amplification.