Where am I?
Mathematicians love to make generalizations. Personally I don’t enjoy that either, but generalizations are very useful in case you’d like to make some mathematical magic. Is there anything cooler than magic?!
Let’s assume (beware of the generalization that is coming towards you) that we had taken N random steps in one dimension. Even though it is very small there still is a possibility that those steps could well be taken into the right hand side (or left). That would have meant that after N random steps, we are standing at +N (or –N). In this case we understand that we are N steps away from the starting point.
If we take half of the N random steps to the right, and other half to the left hand side we would be standing right on the starting point. In that case we would be 0 steps away from the starting point.
These two scenarios are the furthest (N steps) and closest (0 steps) destinations to the starting point after N random steps are taken. Thus, we are finally aware of the fact that after N random steps in one dimension, we have to stop at 0 to N steps away from the starting point.
Q: Is there an algorithm to find out how far we would be to our starting point even before we take a certain number of random steps in one dimension?
For N random steps the answer is the square root of N. For instance if we take 100 random steps in one dimension, we would be √100 = +/- 10 steps away from the starting point.
Click here to learn why it is so.
Now you are wondering: Where and how can we use this information in life?
Place of The Basketball Team
There are 16 teams participating in the Euroleague, which is the most prestigious tournament of Europe. In the regular season of the Euroleague teams get to play with one another twice. In the end of the regular season top 8 teams advance into the playoffs where champion of the season is decided.
Let’s assume that you are supporting a team that is average which means your team would like to fight for the top 8 positions. Just before the season starts you look at the calendar and try to guess how many games your team could win in order to stay in this fight: “If we beat Barcelona at home, and Darüşşafaka on both games…”
You really don’t have to do that. Obviously I will show you how you can use mathematics in order to guess how many wins your team should get.
There are two possible outcomes for a basketball game: Win or lose. It doesn’t matter how strong your opponent is, a game will have two outcomes whatsoever.
Similarities with Random Walks
In one dimension we know that there are two outcomes for a random walk: Right or left. And this is why basketball games can be treated as a random one dimensional walk.
In the regular season each team will play 15×2=30 games. This is same as taking 30 random steps in one dimension.
Then the difference between win and loss column after 30 games can be calculated with taking square root of 30.
√30 = 5,47…
We will call it 6 games. Outcome of these 6 games depend on luck. Your team can win or lose each and every one of them. It proves that after 30 games you either won 6 games more than you lost, or you lost 6 games more than you won:
This information we found using one dimensional random walks tells us that if a team wins between 18 to 12 games in the regular season, that team will be fighting for the playoff positions.
These pictures show how two previous regular seasons ended in the Euroleague. As you can see teams that won between 18 to 12 games fought for a playoff spot.
Try to apply what you learned for one dimensional random walks to a football team that is participating among 18 teams. If it is an average team, what kinds of predictions can you make for the team?
Don’t forget to include the third possibility: Win, lose or draw.
Ps. I didn’t forget about Steve the accountant. We are slowly heading towards the answer.
M. Serkan Kalaycıoğlu