Question 1168570
Let's determine if events A and B are independent or dependent.

**1. Define the Events**

* **Event A:** Rolling a 3 on the first die.
* **Event B:** Getting a sum of more than 6 with the two dice.

**2. Calculate Probabilities**

* **P(A):** The probability of rolling a 3 on the first die is 1/6 (since there's one favorable outcome out of six).
* **P(B):** To find P(B), we need to count the pairs of dice rolls that sum to more than 6.
    * Possible pairs: (1,6), (2,5), (2,6), (3,4), (3,5), (3,6), (4,3), (4,4), (4,5), (4,6), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6).
    * There are 21 pairs that sum to more than 6.
    * Total possible pairs: 6 * 6 = 36.
    * P(B) = 21/36 = 7/12.
* **P(A and B):** The probability of both events A and B occurring.
    * We need to count the pairs where the first die is a 3 and the sum is greater than 6.
    * These pairs are: (3,4), (3,5), (3,6).
    * P(A and B) = 3/36 = 1/12.

**3. Check for Independence**

* Two events are independent if P(A and B) = P(A) * P(B).
* Let's check:
    * P(A) * P(B) = (1/6) * (7/12) = 7/72
    * P(A and B) = 1/12 = 6/72

* Since 7/72 ≠ 6/72, P(A and B) ≠ P(A) * P(B).

**4. Conclusion**

* Therefore, events A and B are **dependent**.