Question

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{11}{16}, p(c)=\frac{1}{3}, p(b|a)=\frac{11}{27}

Explanation:

Step1: Calculate total number of elements in the universal set

The sum of all elements in the Venn - diagram is \(12 + 8+0 + 8+3 + 4+10=45\).

Step2: Calculate \(P(A)\)

The number of elements in \(A\) is \(12 + 8+0 + 8=28\), so \(P(A)=\frac{28}{45}\).

Step3: Calculate \(P(B)\)

The number of elements in \(B\) is \(8 + 8+3+11 = 30\), so \(P(B)=\frac{30}{45}=\frac{2}{3}\).

Step4: Calculate \(P(C)\)

The number of elements in \(C\) is \(0 + 8+3 + 4=15\), so \(P(C)=\frac{15}{45}=\frac{1}{3}\).

Step5: Calculate \(P(A\cap C)\)

The number of elements in \(A\cap C\) is \(0 + 8=8\), so \(P(A\cap C)=\frac{8}{45}\).

Step6: Calculate \(P(B\cap C)\)

The number of elements in \(B\cap C\) is \(8 + 3=11\), so \(P(B\cap C)=\frac{11}{45}\).

Step7: Calculate \(P(B|A)\)

By the formula \(P(B|A)=\frac{P(A\cap B)}{P(A)}\), and \(P(A\cap B)=8 + 8 = 16\), so \(P(B|A)=\frac{16}{28}=\frac{4}{7}\).

Let's check each option:

• Option 1: \(P(A\cap C)=\frac{8}{45}

eq\frac{2}{3}\)

• Option 2: \(P(B\cap C)=\frac{11}{45}\), this is correct.
• Option 3: \(P(A)=\frac{28}{45}

eq\frac{11}{16}\)

• Option 4: \(P(C)=\frac{15}{45}=\frac{1}{3}\), this is correct.
• Option 5: \(P(B|A)=\frac{4}{7}

eq\frac{11}{27}\)

Answer:

B. \(P(B\cap C)=\frac{11}{45}\), D. \(P(C)=\frac{1}{3}\)