Answer:
p=41.79%
Step-by-step explanation:
First let us remember the calculation of a probability:
In this case let's calculate the number of possibilities of winning the tournament when either one of them, Zach(Z) or Shaya(S), wins the first game (number of favorable outcomes). That is:
Given that Z or S won the first game, the player has another 6 possibilities of winning among 12 remaining games, which is a combination:
number of favorable outcomes = 12C6 = 924
Therefore, the winner of the first game has 924 opportunities of winning 6 other games among 12 games and be the tournament winner.
Now, we must find the number of total possible outcomes, which we may analize as it follows:
Assuming that the first winner Z or S loses the tournament we have several outcomes:
1. After winning the first game, the player doesn't win again, then we have 0 possbilities among 7 other games, this is because the next player would win the tournament winning 7 games in a row. We then have the combination 7C0.
2. The player wins just one more game, but doesn't win again. Now we have 1 possibility among 8 games, given that the other player would win the other 7 games. We then have the combination 8C1.
3. The player wins just two more games, but doesn't win again, now we have 2 possibilities among 9 games, given that the other player would win the other 7 games. We then have the combination 9C2.
And so on until we reach the outcome where the player wins 5 more games, but doesn't win again, then we have 5 possibilities among 12 games, given that the other player would win the other 7 games. We then have the combination 12C5.
Now, we must also consider all the outcomes where the first winner wins the tournament, that is the first combination we calculated: 12C6
Therefore we obtain the following:
total number of possible outcomes = 7C0+8C1+9C2+10C3+11C4+12C5+12C6
total number of possible outcomes = 2,211
We have now the elements to calculate the probability:
Hence, the probability that the winner of the first game wins the tournament is 41.79%