use the venn diagram to calculate probabilities. which probability is correct? $p(a|b)=\frac{1}{2}$…

Explanation:

Step1: Recall conditional - probability formula

$P(X|Y)=\frac{P(X\cap Y)}{P(Y)}=\frac{n(X\cap Y)}{n(Y)}$, where $n(X\cap Y)$ is the number of elements in the intersection of $X$ and $Y$, and $n(Y)$ is the number of elements in $Y$.

Step2: Calculate $n(A)$

$n(A)=3 + 1+7 + 6=17$.

Step3: Calculate $n(B)$

$n(B)=1 + 9+6 + 4=20$.

Step4: Calculate $n(C)$

$n(C)=7 + 6+4 + 6=23$.

Step5: Calculate $P(A|B)$

$n(A\cap B)=1 + 6=7$, $P(A|B)=\frac{n(A\cap B)}{n(B)}=\frac{7}{20}\neq\frac{1}{2}$.

Step6: Calculate $P(B|A)$

$n(A\cap B)=7$, $P(B|A)=\frac{n(A\cap B)}{n(A)}=\frac{7}{17}\neq\frac{7}{20}$.

Step7: Calculate $P(A|C)$

$n(A\cap C)=7 + 6=13$, $P(A|C)=\frac{n(A\cap C)}{n(C)}=\frac{13}{23}\neq\frac{6}{23}$.

Step8: Calculate $P(C|A)$

$n(A\cap C)=13$, $P(C|A)=\frac{n(A\cap C)}{n(A)}=\frac{13}{17}$.

Answer:

$P(C|A)=\frac{13}{17}$