Probability problem

[1a2s3d4f5g6h7j8k9l]

Hi!

In the World Series, a team must win 4 out of 7 games. Whichever team wins 4 games first, wins the series. In how many ways can a team win the series?

Please help! Thank you

 

[pka]

In the World Series, a team must win 4 out of 7 games. Whichever team wins 4 games first, wins the series. In how many ways can a team win the series?

There are seven games. Choose how many?

 

[soroban]

Hello, 1a2s3d4f5g6h7j8k9l!

In the World Series, a team must win 4 out of 7 games.
Whichever team wins 4 games first, wins the series.
In how many ways can a team win the series?

Note: when a team has won 4 games, the series is over.

We don't have much choice . . . we must list the outcomes.


\(\displaystyle A\) wins in 4 games:
. . \(\displaystyle AAAA\)
There is \(\displaystyle 1\) way.

\(\displaystyle A\) wins in 5 games: .\(\displaystyle \_\;\_\;\_\;\_\;A\)
. . The first 4 games must have 3 \(\displaystyle A\)'s and 1 \(\displaystyle B.\)
There are: .\(\displaystyle {4\choose3,1} \,=\,4\) ways.


\(\displaystyle A\) wins in 6 games: .\(\displaystyle \_\;\_\;\_\;\_\;\_\;A\)
. . The first 5 games must have 3 \(\displaystyle A\)'s and 2 \(\displaystyle B\)'s.
There are: .\(\displaystyle {5\choose3,2} \,=\,10\) ways.

\(\displaystyle A\) wins in 7 games: .\(\displaystyle \_\;\_\;\_\;\_\;\_\;\_\;A\)
. . The first 6 games must have 3 \(\displaystyle A\)'s and 3 \(\displaystyle B\)'s.
There are: .\(\displaystyle {6\choose3,3} \,=\,20\) ways.

Hence, there are: .\(\displaystyle 1 + 4 + 10 + 20 \:=\:35\) ways that \(\displaystyle A\) can win the series.

. . Similarly, there are \(\displaystyle 35\) ways that \(\displaystyle B\) can win the series.


Therefore, there are \(\displaystyle 70\) ways for either team to win the series.