Question

use the venn diagram to calculate probabilities. which probability is correct? $p(a|b)=\frac{1}{2}$ $p(b|a)=\frac{7}{20}$ $p(a|c)=\frac{6}{23}$ $p(c|a)=\frac{13}{17}$

Explanation:

Step1: Calculate \( P(A|B) \)

By the formula \( P(A|B)=\frac{n(A\cap B)}{n(B)} \). Here \( n(A\cap B) = 1 \), \( n(B)=1 + 9+4=14 \). So \( P(A|B)=\frac{1}{14}
eq\frac{1}{2} \)

Step2: Calculate \( P(B|A) \)

By the formula \( P(B|A)=\frac{n(A\cap B)}{n(A)} \). Here \( n(A\cap B) = 1 \), \( n(A)=3 + 1+7=11 \). So \( P(B|A)=\frac{1}{11}
eq\frac{7}{20} \)

Step3: Calculate \( P(A|C) \)

By the formula \( P(A|C)=\frac{n(A\cap C)}{n(C)} \). Here \( n(A\cap C) = 7 \), \( n(C)=7 + 6+4+6=23 \). So \( P(A|C)=\frac{7}{23}
eq\frac{6}{23} \)

Step4: Calculate \( P(C|A) \)

By the formula \( P(C|A)=\frac{n(A\cap C)}{n(A)} \). Here \( n(A\cap C) = 7 \), \( n(A)=3 + 1+7=11 \). Wait, no. Wait, \( n(A)=3 + 1+7+6=17 \) (if we consider all elements in \( A \) including the intersection with \( C \) and \( B \)). \( n(A\cap C)=7 + 6=13 \). So \( P(C|A)=\frac{13}{17} \)

Answer:

\( P(C|A)=\frac{13}{17} \) (the fourth option)