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.
To find the probability that the Exponents win the series, we need to consider the possible outcomes in which they win at least 3 games.
Let's break it down step by step:
1. The Exponents win the first game (E)
2. The Radicals win the second game (R)
3. The Exponents win the third game (E)
4. The Exponents win the fourth game (R)
5. The Exponents win the fifth game (E)
Now, we need to calculate the probability for each of these outcomes and sum them up.
1. The Exponents win the first game (E):
- Probability of winning at home = 0.60
2. The Radicals win the second game (R):
- Probability of winning on the road = 0.45
3. The Exponents win the third game (E):
- Probability of winning at home = 0.60
4. The Exponents win the fourth game (R):
- Probability of winning on the road = 0.45
5. The Exponents win the fifth game (E):
- Probability of winning at home = 0.60
To calculate the overall probability, we need to multiply the probabilities of each outcome along the schedule:
Probability = (0.60 * 0.45 * 0.60 * 0.45 * 0.60)
Calculating this expression, we get:
Probability ≈ 0.08712
So, the probability that the Exponents win the series is approximately 0.0871, rounded to four decimal places.