If `P(A) = 2/7 and P(B')=1/3 ," then " P(A') + P(B)=`

Options

A. `21/29`
B. `8/29`
C. `29/21`
D. `29/8`

Answer

To solve the problem, we need to find \( P(A') + P(B) \) given \( P(A) = \frac{2}{7} \) and \( P(B') = \frac{1}{3} \).

### Step-by-Step Solution:

1. **Find \( P(A') \)**:
- The probability of the complement of event A, denoted as \( P(A') \), is calculated using the formula:
\[ P(A') = 1 - P(A) \]
- Given \( P(A) = \frac{2}{7} \):
\[ P(A') = 1 - \frac{2}{7} = \frac{7}{7} - \frac{2}{7} = \frac{5}{7} \]

2. **Find \( P(B) \)**:
- The probability of event B can be found from the complement of B, denoted as \( P(B') \):
\[ P(B) = 1 - P(B') \]
- Given \( P(B') = \frac{1}{3} \):
\[ P(B) = 1 - \frac{1}{3} = \frac{3}{3} - \frac{1}{3} = \frac{2}{3} \]

3. **Calculate \( P(A') + P(B) \)**:
- Now that we have both \( P(A') \) and \( P(B) \), we can add them together:
\[ P(A') + P(B) = \frac{5}{7} + \frac{2}{3} \]
- To add these fractions, we need a common denominator. The least common multiple of 7 and 3 is 21.
- Convert each fraction:
\[ \frac{5}{7} = \frac{5 \times 3}{7 \times 3} = \frac{15}{21} \]
\[ \frac{2}{3} = \frac{2 \times 7}{3 \times 7} = \frac{14}{21} \]
- Now add the two fractions:
\[ P(A') + P(B) = \frac{15}{21} + \frac{14}{21} = \frac{29}{21} \]

### Final Answer:
\[ P(A') + P(B) = \frac{29}{21} \]

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