Question

g.pr.10.3 (lc)

which formula is the correct formula to use to test if two events are independent?

\\(p(a \text{ or } b) = p(a) + p(b) - p(a \text{ and } b)\\)

\\(p(a) = \frac{\text{\\# of favorable events}}{\text{total sample space}}\\)

\\(p(a \text{ and } b) = p(a) \cdot p(b)\\)

\\(p(a|b) = p(a \text{ and } b)/p(b)\\)

Explanation:

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<reasoning>

Define mathematical independence for two events

\[
P(A \text{ and } B) = P(A) \cdot P(B)
\]

Evaluate the given options against the definition

\[

$$\begin{aligned} &\text{Option 1: } P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) \quad (\text{Addition Rule for any events})\\ &\text{Option 2: } P(A) = \frac{\text{\# of Favorable events}}{\text{Total sample space}} \quad (\text{Theoretical Probability definition})\\ &\text{Option 3: } P(A \text{ and } B) = P(A) \cdot P(B) \quad (\text{Multiplication Rule for Independent Events})\\ &\text{Option 4: } P(A|B) = \frac{P(A \text{ and } B)}{P(B)} \quad (\text{Conditional Probability definition}) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)\)</mcq-option>
<mcq-option>(B) \(P(A) = \frac{\text{\# of Favorable events}}{\text{Total sample space}}\)</mcq-option>
<mcq-correct>(C) \(P(A \text{ and } B) = P(A) \cdot P(B)\)</mcq-correct>
<mcq-option>(D) \(P(A|B) = P(A \text{ and } B)/P(B)\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Define mathematical independence for two events

\[
P(A \text{ and } B) = P(A) \cdot P(B)
\]

Evaluate the given options against the definition

\[

$$\begin{aligned} &\text{Option 1: } P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) \quad (\text{Addition Rule for any events})\\ &\text{Option 2: } P(A) = \frac{\text{\# of Favorable events}}{\text{Total sample space}} \quad (\text{Theoretical Probability definition})\\ &\text{Option 3: } P(A \text{ and } B) = P(A) \cdot P(B) \quad (\text{Multiplication Rule for Independent Events})\\ &\text{Option 4: } P(A|B) = \frac{P(A \text{ and } B)}{P(B)} \quad (\text{Conditional Probability definition}) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)\)</mcq-option>
<mcq-option>(B) \(P(A) = \frac{\text{\# of Favorable events}}{\text{Total sample space}}\)</mcq-option>
<mcq-correct>(C) \(P(A \text{ and } B) = P(A) \cdot P(B)\)</mcq-correct>
<mcq-option>(D) \(P(A|B) = P(A \text{ and } B)/P(B)\)</mcq-option>
</answer>

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