Two teams are playing in a best of seven playoff series. The first team to win four games wins the series. Ties are broken through sudden decision overtime. If the teams are evenly matched:
What are the odds in favour of either team sweeping the series, in which one team wins four consecutive games??? What are the odds against the series going a full seven games???
Seems like mathhelper gave you a wrong solution. Math helper. WWWWLLL is not possible since WWWW is the end of game. WWWLLLW is the starting point. you can't just go by four same letters since that means the games over. there are 20 possibilities out of 128 since its (1/2)^7 for each, so odds in favor of not going the full 7 games= 11:5
To calculate the odds in favor of one team sweeping the series, we need to determine the probability of that specific outcome occurring. Since the two teams are evenly matched, the probability of one team winning a single game is 0.5.
To win four consecutive games and sweep the series, the team must win the first game, then the second, then the third, and finally the fourth.
The probability of winning four consecutive games can be calculated by multiplying the probabilities of winning each individual game:
P(Sweep) = P(Win 1st game) * P(Win 2nd game) * P(Win 3rd game) * P(Win 4th game)
= 0.5 * 0.5 * 0.5 * 0.5
= 0.0625
So, the odds in favor of one team sweeping the series are 0.0625, or 1 in 16.
Now, to calculate the odds against the series going a full seven games, we need to determine the probability of any other outcome occurring.
The series can end in six games if one team wins four of the first six games. This can happen in three different ways: 4-2, 2-4, or 3-3 with sudden death overtime.
The probability of each outcome is as follows:
P(4-2) = P(Win first 4 games) * P(Lose next 2 games)
= (0.0625) * (0.5 * 0.5)
= 0.015625
P(2-4) = P(Lose first 2 games) * P(Win next 4 games)
= (0.5 * 0.5) * (0.0625)
= 0.015625
P(3-3) = P(Win first 3 games) * P(Lose next 3 games) * P(Sudden decision overtime)
= (0.5 * 0.5 * 0.5) * (0.5 * 0.5 * 0.5) * (0.5)
= 0.03125
Summing up the probabilities of these three outcomes:
P(Series not going 7 games) = P(4-2) + P(2-4) + P(3-3)
= 0.015625 + 0.015625 + 0.03125
= 0.0625
Finally, the odds against the series going a full seven games would be 0.0625, or 1 in 16.
To calculate the odds in favor of one team sweeping the series, you need to consider the possible outcomes. Since one team needs to win four games in a row, there are two ways this can happen: Team A can sweep the series, or Team B can sweep the series.
Let's assume the two teams are evenly matched, so each team has an equal chance of winning any individual game. The probability of one team winning a single game is 1/2, or 0.5.
To calculate the probability of Team A sweeping the series, you need to calculate the probability of Team A winning each of the four games in a row. Since the games are independent events, you multiply the individual probabilities together: 0.5 x 0.5 x 0.5 x 0.5 = 0.0625, or 6.25%.
The same applies to Team B sweeping the series, so the probability is also 6.25%.
Therefore, the combined odds in favor of either team sweeping the series is 6.25% + 6.25% = 12.5%, or 1 in 8 odds.
Now, let's calculate the odds against the series going a full seven games. For the series to end in less than seven games, one team needs to win either 4-0, 4-1, 4-2, or 4-3.
We have already calculated the probability of the series being swept by one team (12.5%). Now, we need to calculate the probability of the series ending in 4-1, 4-2, or 4-3.
For the series to end in 4-1, one team needs to win four games, and the other team needs to win one game. The winning team can be either Team A or Team B, so there are two possibilities. The probability of one team winning four games is 6.25%, and the probability of winning one game is 0.5. Therefore, the probability of the series ending in 4-1 is 2 x (6.25% x 0.5) = 6.25%.
Similarly, for the series to end in 4-2, one team needs to win four games, and the other team needs to win two games. The winning team can be either Team A or Team B, so there are two possibilities. The probability of the series ending in 4-2 is 2 x (6.25% x 0.5 x 0.5) = 3.125%.
For the series to end in 4-3, one team needs to win four games, and the other team needs to win three games. The winning team can be either Team A or Team B, so there are two possibilities. The probability of the series ending in 4-3 is 2 x (6.25% x 0.5 x 0.5 x 0.5) = 1.5625%.
Now, add up the probabilities for all the possibilities of the series ending in fewer than seven games: 12.5% + 6.25% + 3.125% + 1.5625% = 23.4375%.
Therefore, the odds against the series going a full seven games is approximately 1 in 4, or 23.4375%.
To sweep the series a team must have won 4 straight games.
Since the prob of a win is 1/2
the prob of sweep = (1/2)^4 = 1/16
then prob of NOT sweeping the series = 15/16
odds in favor of a sweep = 1 : 15
To go the full 7 games, the outcomes must be WWWWLLL
For the winning team that prob is C(7,4)(1/2)^4 (1/2)^3 = 35/128
consequently the prob of NOT going the 7 games = 93/128
So the odds in favor of not going the full 7 games = 93 : 35