Question

use the venn diagram to calculate probabilities. which probability is correct? \\( p(a) = \frac{3}{5} \\) \\( p(b) = \frac{16}{31} \\) \\( p(a|b) = \frac{2}{7} \\) \\( p(b|a) = \frac{10}{21} \\) (venn diagram with circles a (15, 6), b (10, 6), and u (4) is shown)

Explanation:

Step1: Calculate Total Outcomes

First, find the total number of elements in the universal set \( U \). Sum all regions: \( 15 + 6 + 10 + 4 = 35 \).

Step2: Analyze \( P(A) \)

\( P(A) = \frac{\text{Elements in } A}{\text{Total}} = \frac{15 + 6}{35} = \frac{21}{35} = \frac{3}{5} \)? Wait, no, \( 15 + 6 = 21 \), \( 21/35 = 3/5 \)? Wait, but let's check other options. Wait, maybe I miscalculated. Wait, total is \( 15 + 6 + 10 + 4 = 35 \). Wait, \( P(A) \): elements in A are 15 (only A) + 6 (both A and B) = 21. So \( P(A) = 21/35 = 3/5 \). But let's check other options. Wait, \( P(B) \): elements in B are 6 + 10 = 16. So \( P(B) = 16/35 \), not 16/31. So that's wrong. \( P(A|B) \): conditional probability, \( P(A|B) = \frac{P(A \cap B)}{P(B)} \). \( A \cap B \) is 6, \( P(B) = 16/35 \). So \( 6/(16/35) = 6 * 35/16 = 210/16 = 105/8 = 13.125 \)? No, wait, no: \( P(A|B) = \frac{n(A \cap B)}{n(B)} \), since it's equally likely. So \( n(B) = 6 + 10 = 16 \), \( n(A \cap B) = 6 \), so \( 6/16 = 3/8 \), not 2/7. \( P(B|A) = \frac{n(A \cap B)}{n(A)} \). \( n(A) = 15 + 6 = 21 \), \( n(A \cap B) = 6 \), so \( 6/21 = 2/7 \), not 10/21. Wait, but the first option: \( P(A) = 21/35 = 3/5 \). Wait, but maybe I made a mistake. Wait, total is 15 + 6 + 10 + 4 = 35. So \( P(A) = (15 + 6)/35 = 21/35 = 3/5 \). So that's correct? But wait, the options: first option is \( P(A) = 3/5 \). But let's recheck. Wait, 15 (only A) + 6 (both) = 21. 21/35 = 3/5. Yes. So that's correct? But wait, maybe the problem has a typo, but according to the calculation, \( P(A) = 3/5 \) is correct? Wait, but let's check again. Wait, total elements: 15 (A only) + 6 (A and B) + 10 (B only) + 4 (neither) = 35. So \( P(A) = (15 + 6)/35 = 21/35 = 3/5 \). So that's correct. But wait, the other options: \( P(B) = 16/35 \), not 16/31. \( P(A|B) = 6/16 = 3/8 \), \( P(B|A) = 6/21 = 2/7 \). So the first option \( P(A) = 3/5 \) is correct? Wait, but maybe I messed up. Wait, let's recalculate:

Total elements: 15 (A only) + 6 (A∩B) + 10 (B only) + 4 (neither) = 35.

• \( P(A) \): number of elements in A is 15 + 6 = 21. So \( P(A) = 21/35 = 3/5 \). Correct.
• \( P(B) \): number of elements in B is 6 + 10 = 16. So \( P(B) = 16/35 \), not 16/31. So that's wrong.
• \( P(A|B) \): \( n(A∩B) = 6 \), \( n(B) = 16 \), so \( 6/16 = 3/8 \), not 2/7. Wrong.
• \( P(B|A) \): \( n(A∩B) = 6 \), \( n(A) = 21 \), so \( 6/21 = 2/7 \), not 10/21. Wrong.

So the correct probability is \( P(A) = 3/5 \), so the first option. Wait, but the problem says "which probability is correct". So the first option: \( P(A) = \frac{3}{5} \) is correct.

Answer:

\( P(A) = \frac{3}{5} \) (the first option)