Two teams, the Exponents and the Radicals, square off in a best of 5 math hockey tournament. Once a team wins 3 games, the tournament is over.

The schedule of the tournament (for home games) goes: E-R-E-R-E

If the Exponents are playing at home, there is a 60% chance they'll win. If they are playing on the road, there is a 45% chance they'll win.

Find the probability that the Exponents win the series. Round answers to at least 4 decimal places.

0.0194

your probability seems low

if E wins they can do it with:

0 losses
1 loss
2 losses

for how many combinations can they do each of the above
start there

if E wins, they can do it with:

0 losses out of 3 games: C(2,0) = 1 way to do that, since the last game must be a win

1 loss out of 4 games: C(3,1) = 3 ways to do that, since the last game must be a win

2 losses out of 5 games: C(4,2) = 6 ways to do that, since the last game must be a win

thus, there are 10 total ways to win the tournament, each with its own probability

for example, the probability of winning in 3 games is: 0.6 * 0.45 * 0.6 = 0.162

do that same calculation for each of the 10 ways to win – be careful whether the individual game is home or away – then sum the probabilities across all 10 ways

you should get 0.5756

To find the probability that the Exponents win the series, we need to consider all the possible outcomes of the tournament. Since the first and fourth games are played at the Exponents' home, there are two possible scenarios:

Scenario 1: Exponents win the first game (E), Radicals win the second game (R), Exponents win the third game (E), Exponents win the fourth game (R), Exponents win the fifth game (E).

Scenario 2: Exponents win the first game (E), Radicals win the second game (R), Exponents win the third game (E), Exponents win the fourth game (R), Radicals win the fifth game (E).

For Scenario 1:
- Exponents' chance of winning the first game (E) = 0.60
- Radicals' chance of winning the second game (R) = 0.55 (since it's an away game)
- Exponents' chance of winning the third game (E) = 0.60
- Exponents' chance of winning the fourth game (R) = 0.45
- Exponents' chance of winning the fifth game (E) = 0.60

Therefore, the probability of Scenario 1 occurring is:
0.60 * 0.55 * 0.60 * 0.45 * 0.60 = 0.07134

For Scenario 2:
- Exponents' chance of winning the first game (E) = 0.60
- Radicals' chance of winning the second game (R) = 0.55 (since it's an away game)
- Exponents' chance of winning the third game (E) = 0.60
- Exponents' chance of winning the fourth game (R) = 0.45
- Radicals' chance of winning the fifth game (E) = 0.55 (since it's an away game)

Therefore, the probability of Scenario 2 occurring is:
0.60 * 0.55 * 0.60 * 0.45 * 0.55 = 0.06699

To find the probability that the Exponents win the series, we need to add up the probabilities of all possible outcomes where the Exponents win at least 3 games:

0.07134 + 0.06699 = 0.13833

Therefore, the probability that the Exponents win the series is approximately 0.1383 (rounded to 4 decimal places).