Question

use the venn diagram to calculate conditional probabilities. which conditional probabilities are correct? choose three correct answers. ( p(e|d)=\frac{7}{31} ) ( p(e|f)=\frac{11}{31} ) ( p(f|e)=\frac{1}{11} )

Explanation:

Step1: Calculate total number of elements in \(D\)

Total elements in \(D=13 + 6+1 + 5=25\)
Elements in \(E\cap D = 6 + 1=7\)
By formula \(P(E|D)=\frac{n(E\cap D)}{n(D)}\), \(P(E|D)=\frac{7}{25}\)

Step2: Calculate total number of elements in \(F\)

Total elements in \(F=21 + 7+1 + 5=34\)
Elements in \(E\cap F = 7 + 1=8\)
By formula \(P(E|F)=\frac{n(E\cap F)}{n(F)}\), \(P(E|F)=\frac{8}{34}=\frac{4}{17}\)

Step3: Calculate total number of elements in \(E\)

Total elements in \(E=4 + 6+1 + 7=18\)
Elements in \(F\cap E = 7 + 1=8\)
By formula \(P(F|E)=\frac{n(F\cap E)}{n(E)}\), \(P(F|E)=\frac{8}{18}=\frac{4}{9}\)

Answer:

\(P(E|D)=\frac{7}{25}\) is correct.