Questions: Question 6 of 15 (1 point) Question Attempts: 1 of Unlimited 1 2 3 4 5 6 7 8 9 10 In Salma's bucket there are 9 brown worms and 6 red worms. Salma is going to choose 7 worms at random from the bucket to use for fishing. What is the probability that she will choose 5 brown worms and 2 red worms? Round your answer to three decimal places. Check

Solution Steps

To find the probability that Salma will choose 5 brown worms and 2 red worms, we can use the concept of combinations. First, calculate the number of ways to choose 5 brown worms from 9, and 2 red worms from 6. Then, calculate the total number of ways to choose any 7 worms from the 15 worms in the bucket. The probability is the ratio of the favorable outcomes to the total outcomes.

To find the number of ways to choose 5 brown worms from 9, we use the combination formula:

\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]

Thus, the number of ways to choose 5 brown worms is:

\[ \binom{9}{5} = \frac{9!}{5!(9-5)!} = 126 \]

Next, we calculate the number of ways to choose 2 red worms from 6:

\[ \binom{6}{2} = \frac{6!}{2!(6-2)!} = 15 \]

Now, we find the total number of ways to choose any 7 worms from the total of 15 worms (9 brown + 6 red):

\[ \binom{15}{7} = \frac{15!}{7!(15-7)!} = 6435 \]

The probability \( P \) of choosing 5 brown worms and 2 red worms is given by the ratio of the favorable outcomes to the total outcomes:

\[ P = \frac{\binom{9}{5} \cdot \binom{6}{2}}{\binom{15}{7}} = \frac{126 \cdot 15}{6435} \approx 0.2937062937062937 \]

Rounding this to three decimal places gives:

\[ P \approx 0.294 \]

Final Answer