My friends and I were playing a card game the other night. The game requires each of us to guess whether another is holding a specific card. There are three card possibilities: A, B, and C. There were three rounds. My friend had Card A two rounds in a row. On the third round, we were debating whether he had Card A again.
Option 1: In this round, there is a 1/3 chance that he has Card A.
Option 2: Given that he has received Card A twice in a row, the chance that he will receive it again is unlikely (1/3 * 1/3 * 1/3) = 1/27.
In the absence of any other information, is Option 2 a valid heuristic to guess whether he has Card A? What statistical concept/probabilistic concept does this address? Independence?
Option 1
As any round in your game is completely independent of the others, you shuffle the cards again, they look all the same, the chance of receiving this specific card in each game is 1/3 for each player. Option 2 describes how probable the series of receiving the same card 3 time in a row is within all possible combinations. Note however that any other specific order combination e.g. A,B,C has exactly the same probability. Only when you stop caring about the order and consider A,C,B the same as B,A,C then receiving as three kinds of cards without caring in which order they occur becomes more probable.