Ask a New Question
Menu

You are rolling two number cubes. Let event A

𝐴
be the sum of the rolled numbers is 7 and event B
𝐵
be the first number is 4.

Find the probability P(A and B)
𝑃
(
𝐴

𝑎
𝑛
𝑑

𝐵
)
and determine whether events A
𝐴
and B
𝐵
are independent.

Enter the first answer as a decimal rounded to the nearest hundredth.

(2 points)
​P(A and B)​
​𝑃
(
𝐴

𝑎
𝑛
𝑑

𝐵
)
​
=

Enter 1 if events A
𝐴
and B
𝐵
are dependent. Enter 2 if events A
𝐴
and B
𝐵
are independent. :

1 answer

To find \( P(A \text{ and } B) \) where:

  • Event \( A \): The sum of the rolled numbers is 7.
  • Event \( B \): The first number is 4.

First, let's identify the outcomes for each event.

Step 1: Identify Event B

Event \( B \) occurs when the first die shows 4. The possible outcomes when rolling two number cubes are:

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

So, there are 6 possible outcomes where event \( B \) is true.

Step 2: Identify Event A and A and B

Event \( A \) occurs when the sum of the two dice is 7. The possible pairs of outcomes that give a sum of 7 are:

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

Now, \( A \) and \( B \) occur together (i.e., \( P(A \text{ and } B) \)) when the first die is 4, and the sum of the two dice equals 7. The only pair matching these criteria is:

  • (4, 3)

Thus, \( P(A \text{ and } B) \) corresponds to the event that occurs when the outcomes is (4, 3).

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

There is 1 outcome for \( A \text{ and } B \) which is (4, 3). The total number of outcomes when rolling two dice is \( 6 \times 6 = 36 \). Therefore:

\[ P(A \text{ and } B) = \frac{\text{Number of outcomes of } (A \text{ and } B)}{\text{Total outcomes}} = \frac{1}{36} \approx 0.0278 \]

Rounding to the nearest hundredth, we have:

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

Step 4: Determine Independence of A and B

Events \( A \) and \( B \) are independent if:

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

Calculate \( P(A) \)

There are 6 outcomes where the sum is 7: \[ P(A) = \frac{6}{36} = \frac{1}{6} \]

Calculate \( P(B) \)

There are 6 outcomes where the first die is 4: \[ P(B) = \frac{6}{36} = \frac{1}{6} \]

Calculate \( P(A) \times P(B) \)

\[ P(A) \times P(B) = \left(\frac{1}{6}\right) \times \left(\frac{1}{6}\right) = \frac{1}{36} \]

Step 5: Compare

Since \( P(A \text{ and } B) = \frac{1}{36} \) and \( P(A) \times P(B) = \frac{1}{36} \), we find that:

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

This indicates that events \( A \) and \( B \) are independent.

Final Answers

  • \( P(A \text{ and } B) = 0.03 \)
  • Events \( A \) and \( B \) are independent, so the answer is \( 2 \).

So the final submission would be:

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