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

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

Accepted Answer

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

60%; Yes

75%; No

15%; No

75%; Yes

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

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QUESTION IMAGE

Question

Explanation:

Calculate P(B)

Test for independence

Answer: