Question

in a set of 20, the events for set a and set b are shown in the venn diagram. find p(b). are the events of a and b independent?

60%; yes
75%; no
15%; no
75%; yes

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",
"Venn Diagrams"
]
}
</pre_analysis>

<reasoning>

Calculate P(B)

\[

$$\begin{aligned} &\text{Total sample size } (N) = 20\\ &n(B) = 3 + 12 = 15\\ &P(B) = \frac{n(B)}{N} = \frac{15}{20} = 0.75 = 75\% \end{aligned}$$

\]

Test for independence

\[

$$\begin{aligned} &n(A) = 3 + 3 = 6 \implies P(A) = \frac{6}{20} = 0.30\\ &n(A \cap B) = 3 \implies P(A \cap B) = \frac{3}{20} = 0.15\\ &P(A) \times P(B) = 0.30 \times 0.75 = 0.225\\ &P(A \cap B) = 0.15 eq 0.225 \implies \text{Not independent} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>60%; Yes</mcq-option>
<mcq-correct>75%; No</mcq-correct>
<mcq-option>15%; No</mcq-option>
<mcq-option>75%; Yes</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",
"Venn Diagrams"
]
}
</pre_analysis>

<reasoning>

Calculate P(B)

\[

$$\begin{aligned} &\text{Total sample size } (N) = 20\\ &n(B) = 3 + 12 = 15\\ &P(B) = \frac{n(B)}{N} = \frac{15}{20} = 0.75 = 75\% \end{aligned}$$

\]

Test for independence

\[

$$\begin{aligned} &n(A) = 3 + 3 = 6 \implies P(A) = \frac{6}{20} = 0.30\\ &n(A \cap B) = 3 \implies P(A \cap B) = \frac{3}{20} = 0.15\\ &P(A) \times P(B) = 0.30 \times 0.75 = 0.225\\ &P(A \cap B) = 0.15 eq 0.225 \implies \text{Not independent} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>60%; Yes</mcq-option>
<mcq-correct>75%; No</mcq-correct>
<mcq-option>15%; No</mcq-option>
<mcq-option>75%; Yes</mcq-option>
</answer>

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