a study was conducted on 465 people to find their favorite genre of music. a two-way table below shows the results according to their age. find $p(\text{country} \cup 36-50)$.

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Addition Rule for Probability", "Two-Way Tables" ], "new_concepts": [], "current_concepts": [ "Addition Rule for Probability", "Two-Way Tables" ] } ## Calculate individual probabilities $$ \begin{aligned} &\text{Total sample size } (N) = 465\ &\text{Total Country} = 22 + 30 + 28 + 42 = 122\ &\text{Total 36-50} = 30 + 28 + 38 + 20 = 116\ &P(\text{Country}) = \frac{122}{465}\ &P(36\text{-}50) = \frac{116}{465} \end{aligned} $$ ## Find the intersection $$ \begin{aligned} &\text{Country} \cap 36\text{-}50 = 28\ &P(\text{Country} \cap 36\text{-}50) = \frac{28}{465} \end{aligned} $$ ## Apply the addition rule $$ \begin{aligned} P(\text{Country} \cup 36\text{-}50) &= P(\text{Country}) + P(36\text{-}50) - P(\text{Country} \cap 36\text{-}50)\ &= \frac{122}{465} + \frac{116}{465} - \frac{28}{465}\ &= \frac{210}{465}\ &= \frac{14}{31} \approx 0.4516 \end{aligned} $$ $\frac{14}{31}$ (or approximately $0.4516$) { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Statistics", "Addition Rule for Probability" ] }

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Explanation:

Calculate individual probabilities

Find the intersection

Apply the addition rule

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