Question 1171966: The coach of a college basketball team is in charge of selecting a team for the upcoming season. The coach has hosted try-outs and has cut the possible player list down to 20 players. The coach has to select 12 players from this list of 20 possible players. Suppose that 15 of these 20 players have played at the college level before.
a) Create the probability distribution for the number of experienced players
selected for this team.
b) Comment on the distribution (shape) of the bar graph from part a.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's break down this problem step-by-step.
**a) Probability Distribution for Experienced Players**
* **Variables:**
* N = 20 (Total number of possible players)
* n = 12 (Number of players to be selected)
* K = 15 (Number of experienced players)
* X = Number of experienced players selected (This is our random variable)
* **Distribution:**
* This is a hypergeometric distribution because we are selecting without replacement from a finite population.
* The probability mass function for a hypergeometric distribution is:
```
P(X = x) = [ (K choose x) * (N - K choose n - x) ] / (N choose n)
```
Where:
* (a choose b) = a! / (b! * (a - b)!)
* **Possible Values of X:**
* Since we're selecting 12 players, and there are 15 experienced players, the minimum number of experienced players we could select is 7 (if we select all 5 inexperienced players). The maximum number of experienced players we can select is 12.
* Therefore, X can take values: 7, 8, 9, 10, 11, 12.
* **Calculations:**
* We need to calculate P(X = x) for each value of x.
Let's use a calculator or software to compute the probabilities:
* P(X = 7) = [(15 choose 7) * (5 choose 5)] / (20 choose 12) ≈ 0.0191
* P(X = 8) = [(15 choose 8) * (5 choose 4)] / (20 choose 12) ≈ 0.1147
* P(X = 9) = [(15 choose 9) * (5 choose 3)] / (20 choose 12) ≈ 0.2868
* P(X = 10) = [(15 choose 10) * (5 choose 2)] / (20 choose 12) ≈ 0.3442
* P(X = 11) = [(15 choose 11) * (5 choose 1)] / (20 choose 12) ≈ 0.2065
* P(X = 12) = [(15 choose 12) * (5 choose 0)] / (20 choose 12) ≈ 0.0287
* **Probability Distribution Table:**
| X (Experienced Players) | P(X) |
| :---------------------- | :--------- |
| 7 | 0.0191 |
| 8 | 0.1147 |
| 9 | 0.2868 |
| 10 | 0.3442 |
| 11 | 0.2065 |
| 12 | 0.0287 |
**b) Comment on the Distribution (Shape) of the Bar Graph**
* **Shape:**
* The distribution is unimodal (has one peak) and skewed left.
* The peak of the distribution is at X = 10, indicating that selecting 10 experienced players is the most likely outcome.
* The left skew means that there's a longer tail on the left side, indicating that selecting fewer experienced players (7 or 8) is less likely but still possible.
* Since there are many more experienced players than non experienced players, it is much more likely that the number of experienced players selected will be high.
* **Reasoning:**
* The left skew is due to the fact that there are more experienced players (15) than inexperienced players (5). This makes it more probable to select a higher number of experienced players.
* The highest probability is centered around 10, this is because 10 is the expected value of the distribution.
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