To solve the problem step by step, we will follow the given information and calculate the required probability of choosing a ball that is a multiple of 2 or 3.
### Step 1: Determine the total number of balls
We know the probabilities of choosing white and green balls:
- Probability of white ball = \( \frac{4}{15} \)
- Probability of green ball = \( \frac{2}{5} \)
First, we need to convert the probability of the green ball to a fraction with a common denominator:
- \( \frac{2}{5} = \frac{6}{15} \)
Now, we can find the probability of choosing a blue ball:
\[
P(\text{Blue}) = 1 - P(\text{White}) - P(\text{Green}) = 1 - \frac{4}{15} - \frac{6}{15} = 1 - \frac{10}{15} = \frac{5}{15} = \frac{1}{3}
\]
### Step 2: Calculate the total number of balls
Let \( N \) be the total number of balls in the box. We know there are 10 blue balls, and the probability of choosing a blue ball is \( \frac{1}{3} \):
\[
\frac{10}{N} = \frac{1}{3}
\]
Cross-multiplying gives:
\[
10 \cdot 3 = 1 \cdot N \implies N = 30
\]
### Step 3: Identify the multiples of 2 and 3 from 1 to 30
Now, we will find the multiples of 2 and 3 within the range of 1 to 30.
**Multiples of 2:**
The multiples of 2 from 1 to 30 are:
2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30
Count: 15 multiples of 2.
**Multiples of 3:**
The multiples of 3 from 1 to 30 are:
3, 6, 9, 12, 15, 18, 21, 24, 27, 30
Count: 10 multiples of 3.
### Step 4: Identify the common multiples (multiples of 6)
Now, we need to find the common multiples of 2 and 3, which are the multiples of 6:
The multiples of 6 from 1 to 30 are:
6, 12, 18, 24, 30
Count: 5 multiples of 6.
### Step 5: Use the principle of inclusion-exclusion
To find the total number of favorable outcomes (multiples of 2 or 3), we use the principle of inclusion-exclusion:
\[
\text{Total} = (\text{Multiples of 2}) + (\text{Multiples of 3}) - (\text{Multiples of 6})
\]
\[
\text{Total} = 15 + 10 - 5 = 20
\]
### Step 6: Calculate the probability
The probability of choosing a ball that is a multiple of 2 or 3 is given by:
\[
P(\text{Multiple of 2 or 3}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of balls}} = \frac{20}{30} = \frac{2}{3}
\]
### Final Answer
Thus, the probability of choosing a ball that is a multiple of 2 or 3 is \( \frac{2}{3} \).
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