We are three friends: A, K and S. We were playing a game of basketball free throws with the following rules:
- Every player takes a turn in shooting from locations 1 to 7 (as shown in the attached image) starting from 1. A goes first, then K and then S.
- If the player scores from position 1, he can move to the next position 2 and so on.
- If he doesn't score, the next player takes his turn and so on.
- Whoever scores first from position 7 wins the game.
Pretty easy right?
The problem occurred at some point in the game when after player S failed to score at point 5, player A forgot that it was his turn and K took his turn. So, as a compensation, A was given two chances in his next throw. Eventually, K won the game. If K had not taken the turn of A, probably she(player K) wouldn't have won. So, K argued that the outcome would be the same as A was given two turns later because he lost one of his usual turns. But being A, I think that K had increased her probability of winning because she took my turn and went on to score 2 consecutive throws and eventually win.
The question is that : Does the change of order affect the probability of winning?
Edited: If positions matter to know the probability, here is what exactly happened: K was on 5 and A on 6. K cleared 5 and 6 by taking A's turn and went to position 7, but couldn't score at 7. In the next round A cleared 6 and couldn't clear 7 even though he had two chances. But in the coming round K scored and won the game. All this while, S was in position 4.
If you assume the probability of a player making a shot from a certain position does not depend on the order of play, the only impact would be if K won on the turn taken in error or S won on the turn immediately after. In those cases, A has been deprived of a turn that might have won. Once A got the second chance each player has had the proper number of turns.
For example, suppose the order of play is supposed to be AKSAKSAKSAKSAKS... By mistake, the order of play was AKSAKSKSAAKSAKSAKS... where K and S took their third turns before A. After A's fourth turn everybody has had the proper number of turns and has the expected winning chance as long as the game continues past this point.