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.