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 find \( P(A \text{ and } B) \), we first need to understand what events A and B mean in the context of rolling two dice.

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

First, we can list the pairs of rolls that would satisfy event A (sum = 7):

- (1, 6)
- (2, 5)
- (3, 4)
- (4, 3)
- (5, 2)
- (6, 1)

So, there are 6 outcomes that satisfy event A.

Next, we need to consider when both events A and B happen simultaneously (A and B). For event B to occur (where the first die shows a 4), we only need to look for the outcomes where the first die is 4 and the total sum is 7. The only roll that satisfies this is:

- (4, 3) (the second die must be 3 to make the total 7)

So, there is only 1 outcome that satisfies both events A and B.

Now, we calculate the probability:

There are a total of \(6 \times 6 = 36\) possible outcomes when rolling two number cubes.

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

\[
P(A \text{ and } B) = \frac{\text{Number of outcomes that satisfy both A and B}}{\text{Total number of outcomes}} = \frac{1}{36} \approx 0.02778
\]

Rounding to the nearest hundredth gives:

\[
P(A \text{ and } B) \approx 0.03
\]

Next, we check whether events A and B are independent. To determine independence, we need to check if:

\[
P(A \text{ and } B) = P(A) \cdot P(B)
\]

First, let’s calculate \( P(A) \) and \( P(B) \):

- \( P(A) \): The total outcomes for event A (sum = 7) is 6. So,
\[
P(A) = \frac{6}{36} = \frac{1}{6}
\]

- \( P(B) \): The first die shows 4. There are 6 possible outcomes for the second die, so:
\[
P(B) = \frac{6}{36} = \frac{1}{6}
\]

Now we compute \( P(A) \cdot P(B) \):
\[
P(A) \cdot P(B) = \left(\frac{1}{6}\right) \cdot \left(\frac{1}{6}\right) = \frac{1}{36}
\]

Since \( P(A \text{ and } B) = \frac{1}{36} \) and \( P(A) \cdot P(B) = \frac{1}{36} \), we conclude that the events A and B are independent.

Therefore, the final answers are:

1. \( P(A \text{ and } B) = 0.03 \)
2. Events A and B are independent: enter **2**.