# Two "Heads" Are Indeed Better Than One

**Discrete Mathematics**Level 5

I toss a fair coin 100 times in a row. If the variance of the number of times that two Heads are tossed consecutively is equal to \(\dfrac{a}{b}\), where \(a\) and \(b\) are coprime positive integers, what is the value of \(a+b\)?

**Clarification**: A run of three or more Heads in a row will contribute more than one instance of consecutive Heads. If three Heads are tossed in a row, then two Heads have been tossed consecutively twice. If six Heads are tossed in a row, then two Heads have been tossed consecutively five times, and so on. For example, two Heads have been tossed consecutively 11 times in the following sequence of 30 tosses:

\[ THHTHHHHTHHHTHTHHHHHTTTTHHTHTT \]