Question 1201611: What NCAA college basketball conferences have the higher probability of having a
team play in college basketball’s national championship game? Over the last 20 years,
the Atlantic Coast Conference (ACC) ranks first by having a team in the championship
game 10 times. The South-eastern Conference (SEC) ranks second by having a team in
the championship game 8 times. However, these two conferences have both had teams
in the championship game only one time, when Arkansas (SEC) beat Duke (ACC) 76 - 70 in 1994 (NCAA website, April 2009). Use these data to estimate the following
probabilities.
a) What is the probability the ACC will have a team in the championship game?
b) What is the probability the SEC will have team in the championship game?
c) What is the probability the ACC and SEC will both have teams in the championship
game?
d) What is the probability at least one team from these two conferences will be in the
championship game? That is, what is the probability a team from the ACC or SEC
will play in the championship game?
e) What is the probability that the championship game will not a have team from one
of these two conferences?
Answer by asinus(45)   (Show Source):
You can put this solution on YOUR website! To estimate the probabilities based on the data provided, we will use the following information:
- **Atlantic Coast Conference (ACC)**: 10 appearances in the championship game over 20 years.
- **Southeastern Conference (SEC)**: 8 appearances in the championship game over 20 years.
- **Both conferences had a team in the championship game together only once** (1994).
### Step 1: Calculate the Probabilities
**Total Years**: 20
#### a) Probability the ACC will have a team in the championship game
The probability $ P(\text{ACC}) $ is calculated as:
$$
P(\text{ACC}) = \frac{\text{Number of times ACC had a team}}{\text{Total years}} = \frac{10}{20} = 0.5
$$
#### b) Probability the SEC will have a team in the championship game
The probability $ P(\text{SEC}) $ is calculated as:
$$
P(\text{SEC}) = \frac{\text{Number of times SEC had a team}}{\text{Total years}} = \frac{8}{20} = 0.4
$$
#### c) Probability the ACC and SEC will both have teams in the championship game
Since both conferences had a team in the championship game together only once, the probability $ P(\text{ACC} \cap \text{SEC}) $ is:
$$
P(\text{ACC} \cap \text{SEC}) = \frac{1}{20} = 0.05
$$
#### d) Probability at least one team from these two conferences will be in the championship game
To find the probability that at least one team from the ACC or SEC will be in the championship game, we can use the formula for the union of two events:
$$
P(\text{ACC} \cup \text{SEC}) = P(\text{ACC}) + P(\text{SEC}) - P(\text{ACC} \cap \text{SEC})
$$
Substituting the values we calculated:
$$
P(\text{ACC} \cup \text{SEC}) = 0.5 + 0.4 - 0.05 = 0.85
$$
#### e) Probability that the championship game will not have a team from one of these two conferences
The probability that neither the ACC nor the SEC has a team in the championship game is the complement of the probability that at least one of them does:
$$
P(\text{Neither ACC nor SEC}) = 1 - P(\text{ACC} \cup \text{SEC}) = 1 - 0.85 = 0.15
$$
### Summary of Probabilities
- a) $ P(\text{ACC}) = 0.5 $
- b) $ P(\text{SEC}) = 0.4 $
- c) $ P(\text{ACC} \cap \text{SEC}) = 0.05 $
- d) $ P(\text{ACC} \cup \text{SEC}) = 0.85 $
- e) $ P(\text{Neither ACC nor SEC}) = 0.15 $
These probabilities provide a clear picture of the likelihood of teams from the ACC and SEC participating in the NCAA basketball championship game based on historical data.
|
|
|
