Question

permutations and combinations
this quiz will test your knowledge on permutations and combinations.
value: 1
a small pond contains seven catfish and five bluegill. if four fish are caught at random, what is the probability that at least two catfish have been caught?
a. 85%
b. 88%
c. 79%
d. 91%

Explanation:

Step1: Calculate total number of fish

There are \(7 + 5=12\) fish in total. The number of ways to choose 4 fish out of 12 is given by the combination formula \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n = 12\) and \(r = 4\). So \(C(12,4)=\frac{12!}{4!(12 - 4)!}=\frac{12\times11\times10\times9}{4\times3\times2\times1}=495\).

Step2: Calculate cases of 0 and 1 catfish

Case 0 catfish: Choose 4 blue - gill out of 5. \(C(5,4)=\frac{5!}{4!(5 - 4)!}=5\).
Case 1 catfish: Choose 1 catfish out of 7 and 3 blue - gill out of 5. Using the multiplication principle, the number of ways is \(C(7,1)\times C(5,3)=7\times\frac{5!}{3!(5 - 3)!}=7\times10 = 70\).

Step3: Calculate probability of 0 or 1 catfish

The number of ways to get 0 or 1 catfish is \(5 + 70=75\). The probability of getting 0 or 1 catfish is \(P(0\text{ or }1)=\frac{75}{495}\approx0.1515\).

Step4: Calculate probability of at least 2 catfish

The probability of at least 2 catfish is \(P(\text{at least }2)=1 - P(0\text{ or }1)=1-\frac{75}{495}=\frac{495 - 75}{495}=\frac{420}{495}\approx0.85\) or \(85\%\).

Answer:

A. 85%