https://r2.erweima.ai/mathbot/image/1726656421104bo0yyn0c.jpg

The problem provided involves rolling two dice, with outcomes represented as ordered pairs \((x, y)\), where \(x\) and \(y\) take values from 1 to 6. The task is to calculate the value of \( K \), which represents the probability of the union of two sets \( A \) and \( B \), over the total possible outcomes.

Let's break it down step by step:

### 1. **Understanding the universe \( U \):**
   - The universe \( U \) consists of all possible outcomes when rolling two dice. Since each die can take a value from 1 to 6, the total number of outcomes is \( 6 \times 6 = 36 \).

### 2. **Set \( A \):**
   - \( A = \{(x, y) \in U \mid x + y \leq 4 \} \)
   - We need to find all pairs \((x, y)\) such that the sum \( x + y \leq 4 \).

   These pairs are:
   - \( (1, 1) \), \( (1, 2) \), \( (1, 3) \)
   - \( (2, 1) \), \( (2, 2) \)
   - \( (3, 1) \)
   
   So, \( n(A) = 6 \) (6 elements in set \( A \)).

### 3. **Set \( B \):**
   - \( B = \{(x, y) \in U \mid 4 < x + y \leq 6 \} \)
   - We need to find all pairs \((x, y)\) such that \( 4 < x + y \leq 6 \).

   These pairs are:
   - \( (1, 4) \), \( (1, 5) \)
   - \( (2, 3) \), \( (2, 4) \)
   - \( (3, 2) \), \( (3, 3) \)
   - \( (4, 1) \)
   
   So, \( n(B) = 7 \) (7 elements in set \( B \)).

### 4. **Union of \( A \) and \( B \):**
   - Now, we compute the union \( A \cup B \). Since there are no common elements between sets \( A \) and \( B \) (no pair appears in both sets), we simply add the number of elements:
   \[
   n(A \cup B) = n(A) + n(B) = 6 + 7 = 13
   \]

### 5. **Calculating \( K \):**
   - \( K \) is the ratio of the number of elements in \( A \cup B \) to the total number of elements in \( U \), i.e.:
   \[
   K = \frac{n(A \cup B)}{n(U)} = \frac{13}{36}
   \]

Thus, the value of \( K \) is \( \frac{13}{36} \).

---

Would you like further details or have any questions about the solution?

Here are 5 related questions:
1. What is the probability of the event \( A \) occurring?
2. How do you calculate the intersection of two sets in probability theory?
3. What is the probability that the sum of two dice is greater than 10?
4. How does the concept of mutually exclusive events apply here?
5. How can we extend this method to more than two dice?

**Tip**: When calculating the union of two sets in probability, remember to subtract the intersection if there are common elements, as per the inclusion-exclusion principle.

Considere el experimento de lanzar dos dados comunes. Cada resultado del experimento se designa por un par ordenado (x, y), donde x e y pueden tomar valores del 1 al 6. Si U es el universo de todos los resultados posibles del experimento y A = {(x, y) ∈ U / x + y ≤ 4} y B = {(x, y) ∈ U / 4 < x + y ≤ 6}, calcular el valor de K = n(A∪B)/n(U).

Calculate the probability of events A and B occurring in the experiment of rolling two dice. This problem explores key concepts in probability and set theory, including the inclusion-exclusion principle and the calculation of probabilities using ordered pairs.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation