Adam was on the internet researching prices of new and used cars. One website had 13 cars for sale, 9 of which were vans.
If Adam randomly chose to look at 9 of the cars on the website, what is the probability that exactly 7 of the chosen cars are vans?
Write your answer as a decimal rounded to four decimal places.

To find the probability that Adam randomly chooses exactly 7 vans out of 9 cars, we can solve this using the **hypergeometric distribution**. The hypergeometric distribution models the probability of drawing a certain number of successes (in this case, vans) from a finite population without replacement.

The probability mass function of a hypergeometric distribution is given by the formula:

\[
P(X = k) = \frac{\binom{K}{k} \cdot \binom{N-K}{n-k}}{\binom{N}{n}}
\]

Where:
- \( N \) is the total number of cars,
- \( K \) is the total number of vans,
- \( n \) is the number of cars Adam chooses,
- \( k \) is the number of vans Adam wants to choose.

### Step 1: Identify the parameters
- \( N = 13 \) (total number of cars),
- \( K = 9 \) (total number of vans),
- \( n = 9 \) (cars Adam chooses),
- \( k = 7 \) (vans Adam wants to choose).

### Step 2: Calculate the hypergeometric probability
Using the formula:

\[
P(X = 7) = \frac{\binom{9}{7} \cdot \binom{4}{2}}{\binom{13}{9}}
\]

Now, let's calculate this.The probability that Adam randomly chooses exactly 7 vans out of 9 cars is approximately \( 0.3021 \), rounded to four decimal places.

Would you like a more detailed breakdown of the calculations, or do you have any other questions?

Here are 5 related questions to explore further:
1. What is the probability that Adam chooses exactly 8 vans out of 9 cars?
2. What is the probability that Adam chooses no vans out of 9 cars?
3. How would the probability change if there were 15 cars on the website, with 10 vans?
4. Can this scenario be modeled using a different probability distribution?
5. What is the expected number of vans Adam would choose if he randomly selects 9 cars?

**Tip**: The hypergeometric distribution is useful when sampling without replacement, especially when the sample size is a significant portion of the population.

Discover how to calculate the probability that Adam randomly chooses exactly 7 vans out of 9 cars on a website. This problem uses the hypergeometric distribution, a fundamental concept in probability theory. Suitable for high school students and anyone interested in probability calculations.

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