Question

use the venn diagram to calculate probabilities. which probability is correct? p(a|b) = 1/2 p(b|a) = 7/20 p(a|c) = 6/23 p(c|a) = 13/17

Explanation:

1. Recall the formula for conditional - probability:

• The formula for conditional probability is \(P(X|Y)=\frac{P(X\cap Y)}{P(Y)}\), where \(P(X\cap Y)\) is the probability of the intersection of \(X\) and \(Y\), and \(P(Y)\) is the probability of event \(Y\).
• First, find the total number of elements in the sample - space \(U\): \(n(U)=3 + 1+9 + 6+7 + 4+6 + 8=44\).
• For \(P(A|B)\):
• \(n(A\cap B)=1\), \(n(B)=1 + 9+4 + 6 = 20\). Then \(P(A|B)=\frac{n(A\cap B)}{n(B)}=\frac{1}{20}

eq\frac{1}{2}\).

• For \(P(B|A)\):
• \(n(A\cap B)=1\), \(n(A)=3 + 1+7 + 6 = 17\). Then \(P(B|A)=\frac{n(A\cap B)}{n(A)}=\frac{1}{17}

eq\frac{7}{20}\).

• For \(P(A|C)\):
• \(n(A\cap C)=6 + 7=13\), \(n(C)=6 + 7+6 + 4 = 23\). Then \(P(A|C)=\frac{n(A\cap C)}{n(C)}=\frac{13}{23}

eq\frac{6}{23}\).

• For \(P(C|A)\):
• \(n(A\cap C)=6 + 7 = 13\), \(n(A)=3 + 1+7 + 6 = 17\). Then \(P(C|A)=\frac{n(A\cap C)}{n(A)}=\frac{13}{17}\).

Answer:

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