Question

in a jar, there are blue, orange, and yellow marbles. the probability of randomly choosing a blue marble is $\frac{2}{7}$ and the probability of randomly choosing an orange marble is $\frac{1}{3}$. there are 12 blue marbles.

1. how many total marbles are there?
2. how many orange marbles are in the jar?
3. how many yellow marbles are in the jar?
4. what is the probability of randomly choosing a yellow marble from the jar?

Explanation:

Step1: Find the total number of marbles

Let the total number of marbles be \(x\). The probability of choosing a blue marble is \(P(\text{blue})=\frac{2}{7}\), and the number of blue marbles is \(12\). Using the formula \(P = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}\), we have \(\frac{2}{7}=\frac{12}{x}\). Cross - multiply: \(2x = 12\times7\), so \(x=\frac{12\times7}{2}=42\).

Step2: Find the number of orange marbles

The probability of choosing an orange marble is \(P(\text{orange})=\frac{1}{3}\). Using the formula \(P = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}\), let the number of orange marbles be \(y\). Then \(\frac{1}{3}=\frac{y}{42}\). Cross - multiply: \(y=\frac{42\times1}{3}=14\).

Step3: Find the number of yellow marbles

The number of blue marbles \(b = 12\), the number of orange marbles \(o = 14\), and the total number of marbles \(T = 42\). The number of yellow marbles \(y=T-(b + o)=42-(12 + 14)=42 - 26=16\).

Step4: Find the probability of choosing a yellow marble

Using the formula \(P=\frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}\), the number of yellow marbles is \(16\) and the total number of marbles is \(42\). So \(P(\text{yellow})=\frac{16}{42}=\frac{8}{21}\).

Answer:

1. \(42\)
2. \(14\)
3. \(16\)
4. \(\frac{8}{21}\)