The following table lists World Series Lengths for the fifty years from $1926$ to $1975$. Test at the $0.10$ level whether these data are compatible with the model that each World Series game is an independent Bernoulli trial with $p$ = P(AL wins ) = P (NL wins ) = $\frac {1}{2}$
Table Number of Games ,Number of Years. (respectively)
1) $4, 9$
2) $5, 11$
3) $6 , 8$
4) $7, 22$
Attempt:
I just need help on finding the probabilities.
For example , let X denote the lenght of a World Series.
1) $P( X = 4) $= P (AL and NL wins $4$ times ) =$ 2(\frac{1}{2})^2$
2) $P(X = 5) = \binom {n}{5}(\frac{1}{2})^5(1- \frac{1}{2})^{n - 5}$ I don't even know if this is correct.
3) $P(X = 6)$
I need help on finding probabilities when $X = 5$ and $X = 6$. I think I need to use the binomial distribution. I don't know what to do. Can someone please help me? and show the formula and help me understand.
For, $P(X = 7) = 1 - P(X - 4) - P(X - 5) - P(X - 6).$ I need part 2) and 3).
Thank you in advance.
According to http://www.insidescience.org/content/are-7-game-world-series-more-common-expected/681 the respective probabilities of 4, 5, 6, and 7 games are 0.125,0.25,0.3125 and 0.3125, for evenly matched teams (assuming independently played games). Perhaps knowing the answers, you can straighten out your combinatorial methods. [For example, to have a four game series either AL needs four in a row (probability 1/16) or NL does (total of 1/8).]
Your observed frequencies are 9, 11, 8, and 22. A chi-squared goodness-of-fit test using number of games as categories, has chi-squared statistic 7.712 with DF = 3, giving a P-value of 0.05235.
So, at the 10% level, data are not compatible with equally matched teams and independently played games. At the 5% level it's a close call because the 'chi-squared statistic' doesn't have exactly the chi-squared distribution. (I tried some simulations and mostly the P-values are indeed just above 5%.)
Additional data for more recent years are available in the Wikipedia 'world series' article. It might be interesting to see whether you get a different test result for all years, or for later years.