How does one classify an event into kinds when calculating probabilities?

Question

Say a person tosses a coin 10 straight times and he has a special affinity to heads that he states prior to the tosses. It lands heads each time. This may be seen as impressive.

Say another person invents a game where he/she guesses which person in a room of 10 people has a partner or not. Assume that the probability of each person having a partner is 50%. Now suppose that a person guesses that each person has a partner and he turns out to be right. Suppose this game has never been played before.

Now, clearly, the probability in each scenario is the same. However, coins have been tossed before. In fact, in the history of the universe, trillions of coins have been tossed. When considering this wider set into account, it is not at all surprising that atleast once, a series of 10 tosses landed on heads once.

However, there is no wider set in the case of the partner guessing game. Perhaps one can put that game in the set of "guessing games" in general, but the elements in that set wouldn't be as similar to the partner guessing game the same way coins to other coins would.

Which scenario then should be seen as more impressive? One might say "this is subjective". But one can say this about many things that involve classes of objects. For example, what makes a chair a chair? If one sees two different items that resemble a chair, that "resemblance" itself would also be subjective. Is a chair with a broken leg considered a chair?

Answer 1

The suggestion that the partner game has never been played before is serving as a distraction. Each game offers a “chance set up”. Whether we think of the probabilities in each chance set up as due to the probabilistic structure of the set up itself, or a matter of projected long-term relative frequencies, the probabilities aren’t derived from observation.

If you start out by declaring that each person has a 50% chance of having a partner, that probability won’t change with further observation, any more than if I were to do a couple of coin flips now, that would contribute to some enormous store of evidence from which we derive the 50% probability of heads. It doesn’t. We don’t get the probability of heads from a database.

So, the relative impressiveness of the series of N correct, sequential guesses is the same in each case. (It’s 0.5^N.)