Final answer:
A and B are not independent. The probabilities of drawing a green ticket and drawing the number 8 do not satisfy the conditions of independence.
Step-by-step explanation:
A and B are independent if the following conditions are met:
1. P(AB) = P(A)
2. P(B|A) = P(B)
3. P(A AND B) = P(A)P(B)
In this case, A is the event of drawing a green ticket and B is the event of drawing the number 8. To determine if A and B are independent, we need to calculate P(G₁ AND G₂), P(at least one green), and P(G₂|G₁) and compare them with the corresponding probabilities of A and B.
a. To draw a tree diagram:
- The first level represents the colors (green and non-green)
- The second level represents the numbers (8 and non-8)
- The third level represents the tickets
The tree diagram shows two possible outcomes: G₁G₂ (green ticket with number 8) and non-green ticket with non-8 number.
b. P(G₁ AND G₂) = 1/10 (1 outcome out of 10 possible outcomes)
c. P(at least one green) = 3/10 (3 outcomes out of 10 possible outcomes)
d. P(G₂|G₁) = 1/2 (1 outcome out of 2 green outcomes)
A and B are independent if P(AB) = P(A)P(B).
P(G₁ AND G₂) = P(G₁)P(G₂) = 3/10 * 1/10 = 3/100
Since P(G₁ AND G₂) ≠ P(G₁)P(G₂), A and B are not independent.