This is a question I have for a history of math class, but I can't figure it out. I need to use the three method that Pascal and Fermat used on their problem of points, and it doesn't seem to work out. Here's the question and what I did. Can someone tell me if I'm doing it right and can you help me with the rest, please? Thanks.
Question:
In the World Series, each opponent attempts to win 4 of 7 games. In 1989, Oakland had already won the first two games as the San Francisco Giants and the Oakland Athletics were getting ready to play the third game. Then a major earthquake shook the stadium. Play was suspended and there was discussion on how to proceed with the series. One wants to know what part of the winnings should be awarded to Oakland if the series was suspended. Solve the problem in each of the three ways that it would have been solved by Pascal and Fermat. What part of the winning should be awarded to Oakland if they are twice as skilled as San Francisco?
What I did: For one of the methods, you have to write out the alternatives, and I got these: aA, agA, aggA, agggA, gaA, gagA, gaggA, ggaA, ggagA, gggaA, agggG, gggaG, ggagG, gaggG, gggG.
This gives me a ratio of 10:5 for A's to Giants. I don't know if this is right, but it doesn't work with the next method. I think it's right though.
The second method is using Pascal's Triangle to get the ratio. This site tells you how to do it. I get how it works, but that above ration doesn't work with the triangle: Link
The third method has something to do with solving a simpler problem to find the ratio. I have no idea how to do this method.
Thanks for the help guys, I have no idea what I'm doing wrong.
You didn't say we were to assume each team has $\frac 12$ chance of winning each game. Without that, there is no answer. It could be that the A's will always beat the Giants and their chance of winning the series is $1$.
Your $10:5$ for the first method is not correct because it assumes all the results you have listed have equal probability. The longer strings have less probability than the short ones. If you weight them by $\frac 12$ to the power of the length you will get the correct answer.
A simple method is to assume that the full series is played. The Giants win if they win at least four games out of the remaining five. They have ${5 \choose 4}+{5 \choose 5}=6$ ways to do that. The nice thing of this approach is that all the strings are the same length and have the same probability, $2^{-5}=\frac 1{32}$, so the ratio is $26:6$