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.

(1 point)
P(A and B)=
;
Enter 1 if events A
and B
are dependent. Enter 2 if events A
and B
are independent

Answer 1

To solve the problem, we need to first calculate \( P(A \text{ and } B) \), where event \( A \) is defined as the sum of the rolled numbers being 7, and event \( B \) is defined as the first number being 4.

### Step 1: Identify event A and B

- **Event A**: The possible combinations that give a sum of 7 when rolling two dice are:
- (1, 6)
- (2, 5)
- (3, 4)
- (4, 3)
- (5, 2)
- (6, 1)

So, there are 6 outcomes where the sum is 7.

- **Event B**: The combinations where the first die is 4 are:
- (4, 1)
- (4, 2)
- (4, 3)
- (4, 4)
- (4, 5)
- (4, 6)

There are 6 outcomes where the first die is 4.

### Step 2: Find event A and B (intersection)

To find \( P(A \text{ and } B) \), we look for outcomes that satisfy both \( A \) and \( B \).

- The only combination where the first die is 4 and the total sum is 7 is:
- (4, 3)

So, there is only 1 outcome in the intersection of events A and B.

### Step 3: Calculate \( P(A \text{ and } B) \)

The total number of outcomes when rolling two dice is \( 6 \times 6 = 36 \).

Thus, the probability \( P(A \text{ and } B) \) is:

\[
P(A \text{ and } B) = \frac{\text{Number of outcomes in } A \text{ and } B}{\text{Total outcomes}} = \frac{1}{36}
\]

Calculating the decimal:

\[
P(A \text{ and } B) \approx 0.02778 \quad \text{(rounding to nearest hundredth gives: 0.03)}
\]

### Step 4: Determine if events A and B are independent

To check for independence, we need to compare \( P(A \text{ and } B) \) with \( P(A) \times P(B) \).

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

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

- Now, compute \( P(A) \times P(B) \):
\[
P(A) \times P(B) = \frac{1}{6} \times \frac{1}{6} = \frac{1}{36}
\]

### Conclusion on independence

Since \( P(A \text{ and } B) = \frac{1}{36} \) and \( P(A) \times P(B) = \frac{1}{36} \), events A and B are independent.

### Final Answers:
- \( P(A \text{ and } B) = 0.03 \)
- Events A and B are independent (Enter 2).

Thus, the final answers are:
- \( P(A \text{ and } B) = 0.03 \)
- Events A and B are independent: 2