To find [tex]\( P(A \mid B') \)[/tex], we use the definition of conditional probability. The conditional probability [tex]\( P(A \mid B') \)[/tex] is given by:
[tex]\[ P(A \mid B') = \frac{P(A \cap B')}{P(B')} \][/tex]
### Step 1: Calculate [tex]\( P(B') \)[/tex]
Since [tex]\( P(B) = \frac{1}{3} \)[/tex], the probability of the complement of [tex]\( B \)[/tex], denoted as [tex]\( B' \)[/tex], is:
[tex]\[ P(B') = 1 - P(B) = 1 - \frac{1}{3} = \frac{2}{3} \][/tex]
### Step 2: Find [tex]\( P(A \cap B') \)[/tex]
From the problem, we are given that:
[tex]\[ P(A \cap B') = \frac{2}{9} \][/tex]
### Step 3: Plug values into the conditional probability formula
Now, we can substitute the values into the formula:
[tex]\[ P(A \mid B') = \frac{P(A \cap B')}{P(B')} = \frac{\frac{2}{9}}{\frac{2}{3}} \][/tex]
### Step 4: Simplify the fraction
To simplify:
[tex]\[ P(A \mid B') = \frac{\frac{2}{9}}{\frac{2}{3}} = \frac{2}{9} \times \frac{3}{2} = \frac{2 \cdot 3}{9 \cdot 2} = \frac{6}{18} = \frac{1}{3} \][/tex]
Thus, the conditional probability [tex]\( P(A \mid B') \)[/tex] is [tex]\( \frac{1}{3} \)[/tex].
### Answer:
The correct answer is:
A. [tex]\( \frac{1}{3} \)[/tex]
