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Wednesday, August 29, 2018

Classic Probability Applied (or Not)

My wife and I are watching the America's Got Talent results show. Twelve acts performed last night and and the audience voted. Based on those votes (sort of), seven acts get to stay. If we assume equally likely outcomes, every act as a 7/12 chance of going forward.

The first thing they do is pull aside "three acts in danger" for the Dunkin' Save. Out of this group, the audience re-votes to save one. Of the two left, the judges vote to save one. If the judge vote is a tie, then the audience vote from the previous night determines who stays. Either way, two of these three acts get to stay. If we assume equally likely outcomes in that group, then they have a 2/3 chance of going forward.

Since only seven acts go forward, there are five slots for the remaining nine acts. In other words, they have a 5/9 change of going forward.

Let's recap. Before the results show starts, each act as a 7/12 ≈ 0.58 chance of staying. After this first sort, each act is in one of two situations:
* Dunkin' Save where they have a 2/3 ≈ 0.67 chance of staying.
* Still on stage where they have a 5/9 ≈ 0.56 chance of staying.

It appears that acts are unhappy to be in the Dunkin' Save group, but the probabilities suggest otherwise. Which group would you rather be in? Don't answer right away. Think about it.
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Think a little more.
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Ok. Which group? Why?

Does the equally likely outcomes assumption required by classical probability makes any sense? We know that they're not really equally likely because the acts going forward are not randomly selected. But how does this play out.

All twelve acts are ordered on the the viewers' votes and the Dunkin' Save acts are the 6th, 7th, and 8th place. Therefore, if you're in this group you know that you're "on the bubble" with the audience.  The Save acts could be grouped tightly based on votes. A 2/3 probability night be reasonable and it's a little better than what you had when the show started.

What about the other nine acts? Now you know that you're either in the top five votes or you're at the end of the pack. There's no middle ground left. Would you feel better in this group? If you think you did a great job, then you're really confident. If you think you blew your performance, then you think you're done. The 5/9 probability is probably useless in your mind.

Therefore, the Dunkin' Save group might be neither bad nor good. It's just different. Once the Save group is set aside, the remaining acts probably have a good idea where they stand while the Save group is still in suspense.

Note: There is a potential problem in my use of "probability". Consider a fair coin. If I'm about to flip the coin, the probability of a head is 50%. What if I've already flipped the coin but it's hidden under the couch and no one knows what side us facing up? What's the probability that it's a head? Some would say that it's still 50%. Others would say that the coin flip is already done and, therefore, the probability of a head is either 0 or 1. Our lack of knowledge regarding the outcome doesn't change the fact that it's already done.

If you interpret probabilities the second way, then that could change your preference for being in or out of the Dunkin' Save group. Being put in the Save group puts you into an uncertain outcome where probabilities matter. Being out of the Save group means that your outcome has already been determined (it's 0 or 1) even if you don't know what it is yet.