a marketing department of a cable company is doing research in their area to see how many families have cable television at home. they are also comparing this to the age of the head of the household. the venn diagram shows the results below. analyze the data to find the probability a household does not have cable and the head of household is less than or equal to 50 years old.

cable tv
head of household greater than 50 yrs

7%
28%
27%

accessible description

28%
65%
38%
35%

Question

a marketing department of a cable company is doing research in their area to see how many families have cable television at home. they are also comparing this to the age of the head of the household. the venn diagram shows the results below. analyze the data to find the probability a household does not have cable and the head of household is less than or equal to 50 years old.

cable tv
head of household greater than 50 yrs

7%
28%
27%

accessible description

28%
65%
38%
35%

cable tv
head of household greater than 50 yrs

7%
28%
27%

accessible description

28%
65%
38%
35%

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Probability Formulas" ], "new_concepts": [ "Venn Diagram Probability" ], "current_concepts": [ "Probability Formulas", "Venn Diagram Probability" ] } ## Identify the given probabilities from the Venn diagram $$ \begin{aligned} &P(\text{Cable TV only}) = 7\% \ &P(\text{Cable TV} \cap \text{Head } > 50) = 28\% \ &P(\text{Head } > 50 \text{ only}) = 27\% \end{aligned} $$ ## Calculate the probability of the complement region $$ \begin{aligned} &P(\text{Cable TV} \cup \text{Head } > 50) = 7\% + 28\% + 27\% = 62\% \ &P(\text{Neither}) = 100\% - 62\% = 38\% \end{aligned} $$ ## Relate the complement region to the target event $$ \begin{aligned} &\text{Target event: "does not have cable" AND "head of household is } \le 50\text{"} \ &\text{This is equivalent to: } (\text{Cable TV})^c \cap (\text{Head } > 50)^c = (\text{Cable TV} \cup \text{Head } > 50)^c \ &P(\text{Target}) = 38\% \end{aligned} $$

(A) 28%

(B) 65%

(C) 38%

(D) 35%

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Statistics", "Venn Diagram Probability" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the given probabilities from the Venn diagram

Calculate the probability of the complement region

Relate the complement region to the target event

Answer: