Math Musings

Probability theory  is a branch of mathematics concerned with the analysis of random phenomena. The outcome of a random event cannot be determined before it occurs, but it may be any one of several possible outcomes. The actual outcome is considered to be determined by chance. It is increasingly making its way into all branches of mathematics including number theory where at times mathematicians are looking for answers in a statistical sense. Here is an innocuous-sounding probability question that is not as easy as it sounds. What is the expected number of tosses of an unfair coin needed to get two heads in a row (assume probability p of a head)? Same question for three heads in a row and j heads in a row.  Start with a simple case First: What if you only need the expected number of tosses required to get one head? Let N be the number of coin tosses, then we want to find E(N|1H) (expected number of tosses given that you seek only one head). Toss the coin once. Either you get a head