use the venn diagram to calculate probabilities. which probabilities are correct? select two options. p(a|c)…

Explanation:

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

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

Step2: Calculate $P(A)$

$n(A)=12 + 5+8+6=31$, so $P(A)=\frac{n(A)}{n(U)}=\frac{31}{59}$

Step3: Calculate $P(C)$

$n(C)=6 + 8+3+4 = 21$, so $P(C)=\frac{n(C)}{n(U)}=\frac{21}{59}\neq\frac{3}{7}$

Step4: Calculate $P(A|C)$

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

Step5: Calculate $P(C|B)$

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

Step6: Calculate $P(B|A)$

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

Answer:

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