use the venn diagram to calculate probabilities. which probabilities are correct? select two options.

$p(a|c) = \frac{2}{3}$

$p(c|b) = \frac{8}{27}$

$p(a) = \frac{31}{59}$

$p(c) = \frac{3}{7}$

$p(b|a) = \frac{13}{27}$

Question

use the venn diagram to calculate probabilities. which probabilities are correct? select two options.

$p(a|c) = \frac{2}{3}$

$p(c|b) = \frac{8}{27}$

$p(a) = \frac{31}{59}$

$p(c) = \frac{3}{7}$

$p(b|a) = \frac{13}{27}$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Conditional Probability Calculation", "Theoretical Probability" ], "new_concepts": [], "current_concepts": [ "Theoretical Probability", "Conditional Probability Calculation" ] } ## Find total sample size Using the *Theoretical Probability* knowledge point $$ \begin{aligned} N(U) &= 12 + 5 + 11 + 6 + 8 + 3 + 4 + 10\ &= 59 \end{aligned} $$ ## Calculate individual event sizes Using the *Theoretical Probability* knowledge point $$ \begin{aligned} N(A) &= 12 + 5 + 8 + 6 = 31\ N(B) &= 5 + 11 + 8 + 3 = 27\ N(C) &= 6 + 8 + 3 + 4 = 21 \end{aligned} $$ ## Evaluate simple probabilities Using the *Theoretical Probability* knowledge point $$ \begin{aligned} P(A) &= \frac{N(A)}{N(U)} = \frac{31}{59}\ P(C) &= \frac{N(C)}{N(U)} = \frac{21}{59} e \frac{3}{7} \end{aligned} $$ ## Evaluate conditional probabilities Using the *Conditional Probability Calculation* knowledge point $$ \begin{aligned} P(A|C) &= \frac{N(A \cap C)}{N(C)} = \frac{6 + 8}{21} = \frac{14}{21} = \frac{2}{3}\ P(C|B) &= \frac{N(C \cap B)}{N(B)} = \frac{8 + 3}{27} = \frac{11}{27} e \frac{8}{27}\ P(B|A) &= \frac{N(B \cap A)}{N(A)} = \frac{5 + 8}{31} = \frac{13}{31} e \frac{13}{27} \end{aligned} $$ ## Identify correct options Using the *Theoretical Probability* and *Conditional Probability Calculation* knowledge points $$ \begin{aligned} &\text{Correct: } P(A|C) = \frac{2}{3}\ &\text{Correct: } P(A) = \frac{31}{59} \end{aligned} $$

(A) $P(A|C) = \frac{2}{3}$

(B) $P(C|B) = \frac{8}{27}$

(C) $P(A) = \frac{31}{59}$

(D) $P(C) = \frac{3}{7}$

(E) $P(B|A) = \frac{13}{27}$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Statistics", "Conditional Probability Calculation" ] }

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QUESTION IMAGE

Question

Explanation:

Find total sample size

Calculate individual event sizes

Evaluate simple probabilities

Evaluate conditional probabilities

Identify correct options

Answer: