Question

You are rolling two number cubes. Let event A be the sum of the rolled numbers is 7 and event B be the first number is 4. Find the probability P(A and B) and determine whether events A and B are independent. Enter the first answer as a decimal rounded to the nearest hundredth.

Answer 1

To find the probability \( P(A \text{ and } B) \) and determine if events A and B are independent, let's first define the events:

- Event A: The sum of the rolled numbers is 7.
- Event B: The first number is 4.

### Step 1: Calculate \( P(A \text{ and } B) \)

To find \( P(A \text{ and } B) \), we need to determine the scenarios where both events A and B occur simultaneously.

- If the first die shows 4 (for event B), we need the second die to roll a number such that the total sum is 7. This means:
\[
4 + x = 7 \Rightarrow x = 3
\]

So, the only outcome that satisfies both A and B is \( (4, 3) \).

Now, let's count the total number of outcomes when rolling two number cubes. There are:
\[
6 \times 6 = 36 \text{ total outcomes.}
\]

Therefore, there is only 1 favorable outcome for both events:
- \( (4, 3) \)

Now we can calculate the probability:
\[
P(A \text{ and } B) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{1}{36} \approx 0.02778
\]

Rounding this to the nearest hundredth:
\[
P(A \text{ and } B) \approx 0.03
\]

### Step 2: Check Independence of Events A and B

Events A and B are independent if:
\[
P(A \text{ and } B) = P(A) \cdot P(B)
\]

**Calculate \( P(A) \)**:

To calculate \( P(A) \), we find how many outcomes yield a sum of 7:
- The combinations that yield a total of 7 when rolling two dice are:
- \( (1, 6) \)
- \( (2, 5) \)
- \( (3, 4) \)
- \( (4, 3) \)
- \( (5, 2) \)
- \( (6, 1) \)

There are 6 successful outcomes, so:
\[
P(A) = \frac{6}{36} = \frac{1}{6}
\]

**Calculate \( P(B) \)**:

For event B (first number is 4):
- The combinations where the first die shows 4 are:
- \( (4, 1) \)
- \( (4, 2) \)
- \( (4, 3) \)
- \( (4, 4) \)
- \( (4, 5) \)
- \( (4, 6) \)

There are 6 successful outcomes, so:
\[
P(B) = \frac{6}{36} = \frac{1}{6}
\]

**Calculate \( P(A) \cdot P(B) \)**:
\[
P(A) \cdot P(B) = \frac{1}{6} \cdot \frac{1}{6} = \frac{1}{36}
\]

### Conclusion:

Since:
\[
P(A \text{ and } B) = \frac{1}{36} \quad \text{and} \quad P(A) \cdot P(B) = \frac{1}{36}
\]

Events A and B are independent.

Thus, the answers are:
- \( P(A \text{ and } B) \approx 0.03 \)
- Events A and B are independent.