Probability: What Is Expected Value?

Consider the quote below:

If Alice tosses a coin until she sees a head followed by a tail, and Bob tosses a coin until he sees two heads in a row, then on average, Alice will require $A$ tosses while Bob will require $B$ tosses, even though head-tail and head-head have an equal chance of appearing after two coin tosses.

Source: https://www.quantamagazine.org/20160313-mathematicians-discover-prime-conspiracy/