To investigate this question, consider a related probability question. A weighted coin has probability $p$ to land heads and probability $1-p$ to land tails, $0<p<1$. Suppose I start flipping the coin and stop when I flip my first tails. In this scenario $\text{HHT, HHHT,}$ and $\text{T}$ are all possible sequences of flips. How many flips should I expect to make before a tail forces me to stop?

Looking at a typical sequence $\text{HH} \cdots \text{HT}$, the sequence has probability 1 of being at least one flip long (it has to end with a $\text{T}$!) In front of that final $\text{T}$, each preceding $\text{H}$ appears with probability $p$. Adding our probabilities, we see that our sequence of flips has expected length $$1+p+p^2+\cdots = \frac{1}{1-p}{.}^{**}$$

Let’s return to our baseball scenario. If we wait long enough for every team to win at least one championship, our seasons will appear to be (long) stretches of previous winners punctuated by seasons with a new winner. The stretches of time between new winners will likely increase, because once a team wins its first championship, there is one fewer team that can win its first championship. In all, there will be eight seasons with new winners. A typical sequence of baseball seasons might look like.

During span 1, the probability of having a previous winner is $p=22/30$; during span 2, the probability of having a previous winner is $p=23/30$, etc. These give expected span lengths of $30/8$ years, $30/7$ years, etc.

This problem can be easily generalized. A new sports league (MLS??) with $n$ teams is starting up, and each year one team is determined the champion. How long should we expect until every term has won at least one championship? You can apply the same analysis and find that you should expect $$S_n=\sum_{i=1}^{n} \frac {n}{i}\sim n\ln n$$ seasons to pass until each team wins a championship.

**Notes & Further Questions:**

I’d like to analyze the sequence of actual World Series champions and see how this compares to a random sequence. Is the number of World series champions (twenty-two) higher or lower than expected? This complicated by the fact that the pool of baseball teams has not remained fixed over the years.

The formula $S_n$ comes up in a variety of other situations. The $n^{\text{th}}$ prime number $p_n$ is asymptotic to $S_n$. Also, efficient sorting algorithms work in time proportional to $S_n$. Is there a reason for these connections?