Question

g.pr.10.3 (mc)

\\(p(a) = 0.25\\), \\(p(b) = 0.6\\), and \\(p(a \cap b) = 0.15\\). are a and b independent events? why or why not?

yes, \\(p(a) + p(b) = p(a \cap b)\\), therefore they are independent events
yes, \\(0.25 \cdot 0.6 = 0.15\\), which is \\(p(a \cap b)\\), therefore they are independent events
no, \\(p(a) + p(b) \
eq p(a \cap b)\\), therefore they are dependent events
no, \\(0.25 \cdot 0.6 = 0.15\\), therefore they are dependent events

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Independent Events",
"Probability Formulas"
],
"new_concepts": [],
"current_concepts": [
"Independent Events",
"Probability Formulas"
]
}
</pre_analysis>

<reasoning>

Calculate the product of individual probabilities

\[
P(A) \cdot P(B) = 0.25 \cdot 0.6 = 0.15
\]

Compare with the intersection probability

\[
P(A \cap B) = 0.15
\]
\[
P(A) \cdot P(B) = P(A \cap B)
\]

Determine independence

\[
\text{Since } P(A \cap B) = P(A) \cdot P(B), \text{ the events } A \text{ and } B \text{ are independent.}
\]
</reasoning>

<answer>
<mcq-option>Yes, \(P(A) + P(B) = P(A \cap B)\), therefore they are independent events</mcq-option>
<mcq-correct>Yes, \(0.25 \cdot 0.6 = 0.15\), which is \(P(A \cap B)\), therefore they are independent events</mcq-correct>
<mcq-option>No, \(P(A) + P(B)
eq P(A \cap B)\), therefore they are dependent events</mcq-option>
<mcq-option>No, \(0.25 \cdot 0.6 = 0.15\), therefore they are dependent events</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Statistics",
"Independent Events"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Independent Events",
"Probability Formulas"
],
"new_concepts": [],
"current_concepts": [
"Independent Events",
"Probability Formulas"
]
}
</pre_analysis>

<reasoning>

Calculate the product of individual probabilities

\[
P(A) \cdot P(B) = 0.25 \cdot 0.6 = 0.15
\]

Compare with the intersection probability

\[
P(A \cap B) = 0.15
\]
\[
P(A) \cdot P(B) = P(A \cap B)
\]

Determine independence

\[
\text{Since } P(A \cap B) = P(A) \cdot P(B), \text{ the events } A \text{ and } B \text{ are independent.}
\]
</reasoning>

<answer>
<mcq-option>Yes, \(P(A) + P(B) = P(A \cap B)\), therefore they are independent events</mcq-option>
<mcq-correct>Yes, \(0.25 \cdot 0.6 = 0.15\), which is \(P(A \cap B)\), therefore they are independent events</mcq-correct>
<mcq-option>No, \(P(A) + P(B)
eq P(A \cap B)\), therefore they are dependent events</mcq-option>
<mcq-option>No, \(0.25 \cdot 0.6 = 0.15\), therefore they are dependent events</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Statistics",
"Independent Events"
]
}
</post_analysis>