Question

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

Explanation:

Step1: Calculate the total number of elements

Total \(n(U)=12 + 5+11 + 6+8 + 3+4+10=59\)

Step2: Calculate \(P(A)\)

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

Step3: Calculate \(P(C)\)

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

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

\(n(A\cap C)=6 + 8=14\), \(n(C) = 21\). By the formula \(P(A|C)=\frac{n(A\cap C)}{n(C)}=\frac{14}{21}=\frac{2}{3}\)

Step5: Calculate \(P(C|B)\)

\(n(B\cap C)=8 + 3=11\), \(n(B)=5 + 11+8 + 3=27\). By the formula \(P(C|B)=\frac{n(B\cap C)}{n(B)}=\frac{11}{27}
eq\frac{8}{27}\)

Step6: Calculate \(P(B|A)\)

\(n(A\cap B)=5 + 8=13\), \(n(A)=31\). By the formula \(P(B|A)=\frac{n(A\cap B)}{n(A)}=\frac{13}{31}
eq\frac{13}{27}\)

Answer:

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