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

use the venn diagram to calculate probabilities. which probabilities are correct? select two options. $p(a|c)=\frac{2}{3}$ $p(c|b)=\frac{8}{27}$ $p(a)=\frac{31}{59}$ $p(c)=\frac{3}{7}$ $p(b|a)=\frac{13}{27}$

use the venn diagram to calculate probabilities. which probabilities are correct? select two options. $p(a|c)=\frac{2}{3}$ $p(c|b)=\frac{8}{27}$ $p(a)=\frac{31}{59}$ $p(c)=\frac{3}{7}$ $p(b|a)=\frac{13}{27}$

Answer

Explanation:

Step1: Calculate the total number of elements in the universal set

$12 + 5+11 + 6+8 + 3+4+10=59$

Step2: Calculate $P(A)$

The number of elements in $A$ is $12 + 5+6 + 8=31$. So $P(A)=\frac{31}{59}$.

Step3: Calculate $P(C)$

The number of elements in $C$ is $6 + 8+3 + 4=21$. So $P(C)=\frac{21}{59}\neq\frac{3}{7}$.

Step4: Calculate $P(A|C)$

$A\cap C$ has $6 + 8 = 14$ elements. $P(A|C)=\frac{n(A\cap C)}{n(C)}=\frac{14}{21}=\frac{2}{3}$.

Step5: Calculate $P(C|B)$

$B\cap C$ has $8+3 = 11$ elements. $P(C|B)=\frac{n(B\cap C)}{n(B)}=\frac{11}{5 + 11+8+3}=\frac{11}{27}\neq\frac{8}{27}$.

Step6: Calculate $P(B|A)$

$A\cap B$ has $5 + 8=13$ elements. $P(B|A)=\frac{n(A\cap B)}{n(A)}=\frac{13}{31}\neq\frac{13}{27}$.

Answer:

A. $P(A|C)=\frac{2}{3}$ C. $P(A)=\frac{31}{59}$