How Long Is the All-Star Streak?

Fiddler

The Fiddler Basketball Association’s All-Star Game consists of two teams: “East” and “West.” Every year these two teams play a game, each with a 50 percent chance of winning that’s independent of the outcomes of previous years.

Many, many years into the future, you look at the most recent results of the All-Star Game. On average, what is the longest current winning streak that one of the teams is on? (Here, having won only the most recent game still counts as a “streak” of one game.)

Solution

I will use induction to show the average longest winning streak for $N$ years is

$$M_N = \dfrac{1}{2^N}\sum \limits_{n=1}^N 2^n$$

Year $1$.

There are two outcomes, $E$ and $W$. Both have max streak length of $1$.

$$M_1 = \dfrac{2^1}{2^1} = 1$$

Year $k$.

Assume true.

$$M_k = \dfrac{1}{2^k}\sum \limits_{n=1}^k 2^n$$

Year $k+1$.

For each streak in Year $k$ one of two things can happen:

1. The previous streak holder wins and increases their streak by $1$.
2. The previous streak holder loses and the current longest streak becomes $1$ for the other team.

The total sum of all max streaks thus increases by $2^{k+1}$.

\begin{align*} M_{k+1} &= \dfrac{2^k \cdot M_k + 2^{k+1}}{2^{k+1}}\\ &= \dfrac{\sum \limits_{n=1}^k 2^n + 2^{k+1}}{2^{k+1}}\\ &= \dfrac{\sum \limits_{n=1}^{k+1} 2^n}{2^{k+1}}\\ &= \dfrac{2^{k+2}-2}{2^{k+1}} \end{align*}

Answer

The answer is

$$\boxed{2}$$

Extra Credit

To spice up the All-Star Game, the commissioner of the FBA has decided that there will now be three teams competing in All-Star Games: “Stars,” “Stripes,” and “International.” Each year, two of the three teams play each other. If one year has Stars vs. Stripes, the next year has Stripes vs. International, the year after that has International vs. Stars, and then the cycle repeats with Stars vs. Stripes.

Many, many years after this new format has been adopted, you look at the most recent results of the All-Star game. On average, what is the longest current winning streak that one of the teams is on? (As before, having won one game counts as a “streak.” Also, note that the team with the longest winning streak might not be one of the two teams that played in the most recent All-Star Game.)

Solution

Let $[s, t, i]$ represent the streak of wins in an outcome for “Stars,” “Stripes,” and “International”.

I wrote code that did the following :

1. Initialize the outcomes as $[[0, 0, 0]]$
2. Stars vs Stripes:
   • For each of outcome $[s, t, i]$, two new outcomes are created:
     • Stars could win $\implies [s+1, 0, i]$.
     • Stripes could win $\implies [0, t+1, i]$.
3. Stars vs International:
   • For each of outcome $[s, t, i]$, two new outcomes are created:
     • Stars could win $\implies [s+1, t, 0]$.
     • International could win $\implies [0, t, i+1]$.
4. Stripes vs International:
   • For each of outcome $[s, t, i]$, two new outcomes are created:
     • Stripes could win $\implies [s, t+1, 0]$.
     • International could win $\implies [s, 0, i+1]$.
5. Initialize a total as $0$.
6. For each of outcome $[s, t, i]$, determine the max value and add to the total.
7. Divide the total by the total number of outcomes, which is $2^{\text{years}}$.

Answer

The answer quickly converges to

$$\boxed{2.5}$$

The average streak for each team is $2$, from Fiddler above. It seems to me that half the time the longest streak holder increases by 1, and half the time the longest streak is an average of 2, for the team that is not playing.

This would translate to
$$M_{k+1} = \dfrac{2^k \cdot M_k + 2^k + 2^{k+1}}{2^{k+1}}$$
which would converge to $2.5$, but I did not come up with a clean proof.

Rohan Lewis

2026.02.09

Code can be found here.