2024 AMC 10A Problems/Problem 17
Contents
[hide]Problem
Two teams are in a best-two-out-of-three playoff: the teams will play at most games, and the winner of the playoff is the first team to win games. The first game is played on Team A's home field, and the remaining games are played on Team B's home field. Team A has a chance of winning at home, and its probability of winning when playing away from home is . Outcomes of the games are independent. The probability that Team A wins the playoff is . Then can be written in the form , where and are positive integers. What is ?
Solution
We only have three cases where A wins: AA, ABA, and BAA (A denotes a team A win and B denotes a team B win). Knowing this, we can sum up the probability of each case. Thus the total probability is . Multiplying both sides by 6 yields , so and we find that . Luckily, we know that the answer should contain , so the solution is and the answer is .
~eevee9406
Another way to see the answer is subtraction and not addition is to realize that is between and since it is a probability. ~andliu766
Video Solution 1 by Pi Academy
https://youtu.be/fW7OGWee31c?si=oq7toGPh2QaksLHE
Video Solution2 by SpreadTheMathLove
https://www.youtube.com/watch?v=6SQ74nt3ynw
See also
2024 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 16 |
Followed by Problem 18 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
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