To solve the problem of finding the probability that \( x_1 + x_2 + x_3 \) is odd, we will follow these steps:
### Step 1: Identify the contents of each box
- Box 1 contains cards: {1, 2, 3}
- Box 2 contains cards: {1, 2, 3, 4, 5}
- Box 3 contains cards: {1, 2, 3, 4, 5, 6, 7}
### Step 2: Determine the probabilities of drawing odd and even numbers from each box
- **Box 1**:
- Odd numbers: {1, 3} → 2 odd numbers
- Even numbers: {2} → 1 even number
- Probability of odd \( P(O_1) = \frac{2}{3} \)
- Probability of even \( P(E_1) = \frac{1}{3} \)
- **Box 2**:
- Odd numbers: {1, 3, 5} → 3 odd numbers
- Even numbers: {2, 4} → 2 even numbers
- Probability of odd \( P(O_2) = \frac{3}{5} \)
- Probability of even \( P(E_2) = \frac{2}{5} \)
- **Box 3**:
- Odd numbers: {1, 3, 5, 7} → 4 odd numbers
- Even numbers: {2, 4, 6} → 3 even numbers
- Probability of odd \( P(O_3) = \frac{4}{7} \)
- Probability of even \( P(E_3) = \frac{3}{7} \)
### Step 3: Determine the cases for \( x_1 + x_2 + x_3 \) to be odd
For the sum \( x_1 + x_2 + x_3 \) to be odd, we can have the following combinations:
1. One odd and two even (OEE)
2. Three odd (OOO)
### Step 4: Calculate the probability for each case
#### Case 1: One odd and two even (OEE)
- **OEE Combinations**:
1. \( O_1, E_2, E_3 \)
2. \( E_1, O_2, E_3 \)
3. \( E_1, E_2, O_3 \)
Calculating the probabilities for each combination:
1. \( P(O_1) \cdot P(E_2) \cdot P(E_3) = \frac{2}{3} \cdot \frac{2}{5} \cdot \frac{3}{7} = \frac{12}{105} \)
2. \( P(E_1) \cdot P(O_2) \cdot P(E_3) = \frac{1}{3} \cdot \frac{3}{5} \cdot \frac{3}{7} = \frac{9}{105} \)
3. \( P(E_1) \cdot P(E_2) \cdot P(O_3) = \frac{1}{3} \cdot \frac{2}{5} \cdot \frac{4}{7} = \frac{8}{105} \)
Adding these probabilities together:
\[
P(OEE) = \frac{12}{105} + \frac{9}{105} + \frac{8}{105} = \frac{29}{105}
\]
#### Case 2: Three odd (OOO)
Calculating the probability:
\[
P(O_1) \cdot P(O_2) \cdot P(O_3) = \frac{2}{3} \cdot \frac{3}{5} \cdot \frac{4}{7} = \frac{24}{105}
\]
### Step 5: Combine the probabilities from both cases
Total probability that \( x_1 + x_2 + x_3 \) is odd:
\[
P(\text{odd}) = P(OEE) + P(OOO) = \frac{29}{105} + \frac{24}{105} = \frac{53}{105}
\]
### Final Answer
The probability that \( x_1 + x_2 + x_3 \) is odd is \( \frac{53}{105} \).
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