I was playing a game using $7$ cards (one for each player), and there were $2$ aces in those cards. Each person would draw a card every round, and and then we would give them back for another shuffle (replacement). In the first two rounds, I got an ace twice, and in the third round I said that there would be a very low chance for me to get an ace for the third time in a row. However, someone said "You are just as likely to get an ace as me". I understand that that person was considering the probability
$$P(\text{3rd ace | 2 aces in a row}) = \frac{2} {7}, $$
while I was considering
$$P(\text{getting 3 aces in a row}) = \left(\frac{2} {7}\right) ^3. $$
My question would be who had the right reasoning in the situation: me, him, or are we both right due to the probability space each considers?
Best Answer
I think you have been trapped in the gambler's fallacy!
You shuffle the deck at every round so the probability to fish an ace at the n-th round (after the results of the other rounds are already established) doesn't depend on what already happened earlier.
So , the first probability is the right one, you can see also that :
$$P( \text {n-th ace in the n-th round | n-1 aces in a row }) = P( \text{n-th ace in the n-th round} ) = \frac{2}{7}$$
Instead at the start of the game to fish an ace for $n$ consecutive round is equal to $(\frac{2}{7})^n$. But this just because nothing happened yet!