Today I want to take a quick look at a simple real-life semi-paradox that I observed after looking at NBA Draft Lottery system.
The Paradox
Let's say the following is the weather forecast for tomorrow morning,
- Rain hard : 45%
- Tiny drizzle : 30%
- Cloudy : 20%
- Hot sunny day : 5%
Naively speaking from this data, it is wiser for people to take their umbrella/wear a raincoat to their office. Right?
But is it really true?
At a glance, you can see that the condition 'Rain hard', clearly has the most probability of occuring than the other possibilities. But you may forget to notice that it is more likely that it won't rain cats and dogs tomorrow morning. Yes. The probability of it won't rain hard tomorrow morning is 55%, and it's bigger than 45% !
So now we arrive to the paradox :
It's more likely that it'll be rain hard compared to each of the other weather conditions. But it's more likely that it won't rain hard tomorrow?
Then the probability itself means no good afterall? Even if one got the greatest chance than the others?
Let's now consider a tangible, real-life consequence of this issue.
NBA Draft Lottery
If you are familiar with NBA, then you know that every year NBA means to distribute the best rookies to the worst teams. Between the 14 lowest performing teams, they will get a chance to get the #1 rookie that comes in next season. The order below starts with the worst team, 2nd worst team, etc.
- 250 combinations, 25.0% chance of receiving the #1 pick
- 199 combinations, 19.9% chance
- 156 combinations, 15.6% chance
- 119 combinations, 11.9% chance
- 88 combinations, 8.8% chance
- 63 combinations, 6.3% chance
- 43 combinations, 4.3% chance
- 28 combinations, 2.8% chance
- 17 combinations, 1.7% chance
- 11 combinations, 1.1% chance
- 8 combinations, 0.8% chance
- 7 combinations, 0.7% chance
- 6 combinations, 0.6% chance
- 5 combinations, 0.5% chance
Again, the same dilemma. The worst team will get the highest chance (25%), yet it's more likely for other teams to get the players. And the reality? Below are the data from the last 23 years of NBA rookie drafting on which teams got the opportunity to select the #1 rookie
- 1st : 3 times
- 2nd : 4 times
- 3rd : 5 times
- 5th : 5 times
- 6th : 2 times
- 7th : 1 time
- 8th : 1 time
- 9th : 1 time
- 11th : 1 time
The funny thing is that the worst team only get the #1 rookie 3/23 ~ 13%, which is around half of their expected probability ! But more than anything, this empirical data, although lacking in numbers of experiment, practically confirms that the worst team really got no edge of getting the expectedly, most talented boy than the other teams.
To sum it all, I try to reconstruct some opinions from each party regarding their positions before the lottery
From the worst team's point of view,
We are in the pole position of rookie drafting? It's still much more likely that we didn't get the #1 rookie !
From the other team's point of view
We really wish we are the worst team, they have the greatest chance of getting the #1 rookie !
Intriguing, isn't it?
Reference/Note :
- NBA Draft Lottery data taken from http://en.wikipedia.org/wiki/NBA_Draft_Lottery#Process
- This paradox won't occur in case that the highest party has more than 50% probability (that's why I call it semi-paradox) So can we just all agree that one thing is superior if that they have more than half the chance?
Intriguing indeed =D
ReplyDeleteI have two issues on this matter.
The first one is whether there is a common action among the other possibilities that we can regard the non-highest probabilities as one possibility. Essentially my first point tries to see the paradox from the applications of looking at probabilities: the decision that we want to take based on the numbers. And in some cases, there might be "invisible weight" that we give to each possibility that may cause us to act beyond considering only the highest possibility.
For example in the weather forecast case, You may argue that the choice of not bringing an umbrella is applicable to the 55% not-raining-hard condition, and so in this case you can argue that bringing umbrella is "better" choice in the sense 55% of the time the umbrella won't be so useful. But actually in this case bringing an umbrella is not a big deal, and the cost of having trapped in heavy rain is bigger that bringing umbrella unnecessarily. So although the semi-paradox seems to happen here, actually it's not really the case, as we can see that the probability of convenience, given that we bring an umbrella is most likely higher (and greater than 50%) compared to not bringing an umbrella.
My second point is that, what is this that you are saying "the worst team really got no edge of getting the expectedly, most talented boy than the other teams"? I think it's just because we don't have enough data or there are other factors that cause the probabilities you gave here to be inaccurate, otherwise, law of large numbers tells us that in the long run the team with highest probability will get it most of the time compared to other teams.
Probably you are getting insight on this semi-paradox mostly because of the remarks made by the two teams that seem to be paradoxical in themselves (and it seems paradoxical because here we can't regard the non-highest possibility to be one, since the goal here for each team is to get the #1 rookie), cmiiw. =D