Final answer:
The probability that exactly 4 out of 6 rocks are sedimentary is calculated by using combinations to find the total number of ways to choose 6 rocks, and the specific ways to choose 4 sedimentary rocks. The probability is found to be approximately 0.0416 when rounded to four decimal places.
Step-by-step explanation:
To calculate the probability that exactly 4 out of 6 selected rocks are sedimentary from a shipment that contains 9 igneous, 7 sedimentary, and 7 metamorphic rocks, we can use combinations to determine the various ways to choose the rocks.
First, the total number of ways to select 6 rocks from the total of 9 + 7 + 7 = 23 rocks is calculated using the combination formula C(n, k) = n! / (k!(n - k)!), where '!' signifies factorial:
C(23, 6) = 23! / (6!(23 - 6)!) = 100947
Next, we consider the ways to select 4 sedimentary rocks out of the available 7:
C(7, 4) = 7! / (4!(7 - 4)!) = 35
Then, because we want exactly 4 sedimentary rocks, we need to select 2 more rocks from the remaining non-sedimentary 16 rocks (9 igneous + 7 metamorphic):
C(16, 2) = 16! / (2!(16 - 2)!) = 120
Now, we calculate the number of ways to select 4 sedimentary rocks and 2 non-sedimentary rocks together:
Total ways = C(7, 4) * C(16, 2) = 35 * 120 = 4200
Thus, the probability that exactly 4 out of 6 rocks are sedimentary is the number of successful ways divided by the total number of ways to select 6 rocks:
Probability = Total ways / C(23, 6) = 4200 / 100947 ≈ 0.0416 (rounded to four decimal places)