Question

in teresas bucket there are 5 brown worms and 8 red worms. teresa is going to choose 8 worms at random from the bucket to use for fishing. what is the probability that she will choose 2 brown worms and 6 red worms? round your answer to three decimal places.

Explanation:

Step1: Calculate total number of worms

Total worms = \(5 + 8=13\).

Step2: Use combination formula \(C(n,k)=\frac{n!}{k!(n - k)!}\)

Number of ways to choose \(8\) worms from \(13\): \(C(13,8)=\frac{13!}{8!(13 - 8)!}=\frac{13!}{8!5!}=\frac{13\times12\times11\times10\times9}{5\times4\times3\times2\times1}=1287\).

Step3: Calculate number of ways to choose \(2\) brown and \(6\) red

Number of ways to choose \(2\) brown from \(5\): \(C(5,2)=\frac{5!}{2!(5 - 2)!}=\frac{5\times4}{2\times1}=10\).
Number of ways to choose \(6\) red from \(8\): \(C(8,6)=\frac{8!}{6!(8 - 6)!}=\frac{8\times7}{2\times1}=28\).
Total favorable ways: \(10\times28 = 280\).

Step4: Calculate probability

Probability \(P=\frac{280}{1287}\approx0.218\).

Answer:

\(0.218\)