Problem in realizing the World Series problem

Question

I have a problem in realizing the World Series solution. The World Series is a seven-game series that terminates as soon as either team wins four games. Let X be the random variable that represents the outcome of a World Series between teams A & B; Possible values of X are AAAA, BABABAB and BBBAAA. Let Y be the number of games played, which ranges from 4 to 7. Now we want H(X) and Y(X|Y) . ( H(X) is the entropy of X).

My problem is in the first part of solution. Can anyone explain this part for me:

Answer 1

In order to compute the entropy of $X$, you must enumerate its possible outcomes and the corresponding probabilities. Note that the possible $X$ events correspond to either $Y=4,5,6,$ or $7$. The enumeration goes as follows:

• For $Y=4$, either $X=AAAA$ or $BBBB$. Thats two events of probability $(1/2)^4$ (assuming that both teams have equal probability of winning a game)
• Consider $Y=5$. Either $A$ won the series or $B$. If $A$ won the series, $A$ must have won the last (fifth) game and $3$ out of the $4$ first games, hence there are $4 \choose 3$ possible $X$ events corresponding to $A$ winning the series in $Y=5$ games. The same number of events also holds for the case of $B$ winning the series in $Y=5$ games. Therefore, for this case there are a total of $2 {4 \choose 3}$ events, each with a probability $(1/2)^5$
• I will leave the enumeration of $X$ for $Y=6,7$ to you...