Marcus and Allen are playing Rock/Paper/Scissors. The first one to win 10 times does not have to do the dishes. The score is now 9 to 7 in favor of Marcus. If ties are ignored and not counted, what is the probabilty Marcus will not have to do the dishes?
He only needs to win one more time to watch TV instead of doing dishes.
Assuming he is not a pro, and everything being equal, and since history does not count, what is the probability for him to win the next one?
To find the probability that Marcus will not have to do the dishes, we need to consider the possible outcomes for the remaining games.
Since the score is currently 9 to 7 in favor of Marcus, he needs only one more win to reach 10 wins. Allen, on the other hand, needs three more wins.
Let's analyze the possible scenarios:
1. Marcus wins the next game:
- Marcus wins the next game: Marcus wins and does not have to do the dishes.
- Allen wins the next game: The match continues.
2. Allen wins the next game:
- Marcus wins the next game: The match continues.
- Allen wins the next game: The match continues.
From the above scenarios, we can see that Marcus has two opportunities to win the next game and avoid having to do the dishes. We can calculate the probability of this happening by dividing the number of favorable outcomes (where Marcus wins) by the total number of possible outcomes.
Out of the four possible outcomes, two are favorable for Marcus, and two are not. Therefore, the probability that Marcus will not have to do the dishes is 2/4, which simplifies to 1/2, or 50%.
To find the probability that Marcus will not have to do the dishes, we need to determine the number of possible outcomes and the number of favorable outcomes.
There are two players, Marcus and Allen, and three possible moves each: Rock, Paper, and Scissors. Since ties are not counted, there are two possible outcomes for each round: Marcus wins or Allen wins.
Given that the current score is 9 to 7 in favor of Marcus, we know that there have already been 16 rounds played. To win the game, Marcus needs to win one more round (out of a possible three: rock, paper, or scissors), while Allen needs to win three more rounds (out of a possible four: rock, paper, or scissors, or a tie).
Now, let's calculate the number of favorable outcomes:
For Marcus to win the game, he needs just one more win, which can happen in any of the three possible moves: rock, paper, or scissors.
For Allen to win the game, he needs to win three out of the four remaining rounds. Since there are three possible moves and a tie, the number of ways Allen can win three rounds is given by:
3 × 3 × 3 = 27
Therefore, the number of favorable outcomes is 3 (for Marcus) and 27 (for Allen).
To find the probability that Marcus will not have to do the dishes, we can divide the number of favorable outcomes (3) by the number of possible outcomes (3 + 27 = 30):
P(Marcus does not have to do the dishes) = 3 / 30 = 0.1 or 10%
So, the probability that Marcus will not have to do the dishes is 10%.