GMAT Challenge Series (700+): Q6

Question:

Baseball's World Series matches 2 teams against each other in a best-of-seven series. The first team to win four games wins the series and no subsequent games are played. If you have no special information about either of the teams, what is the probability that the World Series will consist of fewer than 7 games?

A. 12.5%

B. 25%

C. 31.25%

D. 68.75%

E. 75%

$$\text{Assume that each team has a 50% chance of winning each game.}$$ $$\text{Series will go to game 7 if each team has won 3 games from the first 6 games.}$$ $$\text{Example: A, A, A, B, B, B, ?}$$ $$\textit{Probability(7 Games)} = {{\left(\frac{1}{2}\right)^{6}}{6\,!} \over \left({3\,!}\right)\left({3\,!}\right)}$$ $$\textit{Probability(7 Games)} = {{\left(\frac{1}{64}\right)}{720} \over \left({6}\right)\left({6}\right)}$$ $$\textit{Probability(7 Games)} = \frac{20}{64} = \frac{5}{16} = \textit{31.25%}$$ $$\textit{Probability(Fewer than 7 Games)} = {1} - \textit{Probability(7 Games)}$$ $$\textit{Probability(Fewer than 7 Games)} = {1} - \textit{31.25%} = \textit{68.75%}$$