To address the given questions, let's first consider the values provided:
- Probability of event [tex]\( A \)[/tex] occurring, [tex]\( P(A) = 0.12 \)[/tex] or [tex]\( 12\% \)[/tex].
- Probability of event [tex]\( B \)[/tex] occurring, [tex]\( P(B) = 0.15 \)[/tex] or [tex]\( 15\% \)[/tex].
- Probability of both events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] occurring, [tex]\( P(A \cap B) = 0.03 \)[/tex] or [tex]\( 3\% \)[/tex].
### 1. Conditional Probability [tex]\( P(A \mid B) \)[/tex]:
The conditional probability of [tex]\( A \)[/tex] given [tex]\( B \)[/tex] is calculated using the formula:
[tex]\[
P(A \mid B) = \frac{P(A \cap B)}{P(B)}
\][/tex]
Substituting the given values:
[tex]\[
P(A \mid B) = \frac{0.03}{0.15} = 0.2 \text{ or } 20\%
\][/tex]
### 2. Conditional Probability [tex]\( P(B \mid A) \)[/tex]:
The conditional probability of [tex]\( B \)[/tex] given [tex]\( A \)[/tex] is calculated using the formula:
[tex]\[
P(B \mid A) = \frac{P(A \cap B)}{P(A)}
\][/tex]
Substituting the given values:
[tex]\[
P(B \mid A) = \frac{0.03}{0.12} = 0.25 \text{ or } 25\%
\][/tex]
### 3. Independence of Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
Two events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are considered independent if and only if the probability of their intersection is equal to the product of their individual probabilities:
[tex]\[
P(A \cap B) = P(A) \cdot P(B)
\][/tex]
Calculating the product of the individual probabilities:
[tex]\[
P(A) \cdot P(B) = 0.12 \cdot 0.15 = 0.018 \text{ or } 1.8\%
\][/tex]
Comparing this with [tex]\( P(A \cap B) \)[/tex]:
[tex]\[
0.03 \neq 0.018
\][/tex]
Since [tex]\( P(A \cap B) \)[/tex] is not equal to [tex]\( P(A) \cdot P(B) \)[/tex], events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are not independent.
### Final Answer:
Drag-and-drop responses:
- [tex]\( P(A \mid B) = 20\% \)[/tex]
- The probability of [tex]\( A \)[/tex] given [tex]\( B \)[/tex] is not the same as the probability of [tex]\( A \)[/tex], so [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are not independent events.
