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

First, we find the total number of marbles in the bag. The number of yellow marbles is 8, green is 9, purple is 3, and red is 5. So total marbles \( n = 8 + 9 + 3 + 5 = 25 \).

Step2: Define "at most one purple marble"

"At most one purple marble" means either 0 purple marbles or 1 purple marble. We use combinations to calculate the number of ways to choose marbles. The formula for combinations is \( \binom{n}{k}=\frac{n!}{k!(n - k)!} \), where \( n \) is the total number of items, and \( k \) is the number of items to choose.

Step3: Calculate number of ways for 0 purple marbles

If we choose 0 purple marbles, we are choosing 3 marbles from the non - purple marbles. The number of non - purple marbles is \( 25-3 = 22 \). So the number of ways to choose 3 non - purple marbles is \( \binom{22}{3}=\frac{22!}{3!(22 - 3)!}=\frac{22\times21\times20}{3\times2\times1}=1540 \).

Step4: Calculate number of ways for 1 purple marble

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

Step5: Calculate total number of favorable outcomes

The total number of favorable outcomes (at most one purple marble) is the sum of the number of ways to choose 0 purple marbles and 1 purple marble. So \( N=1540 + 693=2233 \).

Step6: Calculate total number of ways to choose 3 marbles

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

Step7: Calculate the probability

The probability \( P \) of choosing at most one purple marble is the number of favorable outcomes divided by the total number of outcomes. So \( P=\frac{2233}{2300}\approx0.971 \).

Answer:

0.971