Question

a bag contains eight yellow marbles, nine green marbles, three purple marbles, and five red marbles. three marbles are randomly chosen from the bag. what is the probability that there is at most one purple marble? 0.100 0.301 0.770 0.971

Explanation:

Step1: Calculate total number of marbles

The total number of marbles is \(8 + 9+3 + 5=25\) marbles.

Step2: Define the cases for at - most one purple marble

The cases are 0 purple marbles and 1 purple marble.

Step3: Calculate the number of ways to choose 3 marbles with 0 purple marbles

The number of non - purple marbles is \(25 - 3=22\). The number of ways to choose 3 non - purple marbles is \(C(22,3)=\frac{22!}{3!(22 - 3)!}=\frac{22\times21\times20}{3\times2\times1}=1540\).

Step4: Calculate the number of ways to choose 3 marbles with 1 purple marble

The number of ways to choose 1 purple marble out of 3 is \(C(3,1)=\frac{3!}{1!(3 - 1)!}=3\). The number of ways to choose 2 non - purple marbles out of 22 is \(C(22,2)=\frac{22!}{2!(22 - 2)!}=\frac{22\times21}{2\times1}=231\). The number of ways to choose 1 purple and 2 non - purple marbles is \(C(3,1)\times C(22,2)=3\times231 = 693\).

Step5: Calculate the total number of favorable cases

The total number of favorable cases (0 or 1 purple marble) is \(1540+693 = 2233\).

Step6: Calculate the total number of ways to choose 3 marbles

The total number of ways to choose 3 marbles out of 25 is \(C(25,3)=\frac{25!}{3!(25 - 3)!}=\frac{25\times24\times23}{3\times2\times1}=2300\).

Step7: Calculate the probability

The probability \(P=\frac{2233}{2300}=0.971\).

Answer:

0.971