https://r2.erweima.ai/mathbot/image/17270435901684ow97fg5.jpg

In the image, we are given two independent events \( A \) and \( B \), with the following information:

\[
P(\bar{A}) = 0.40 \quad \text{and} \quad P(B) = 0.70
\]
Where \( \bar{A} \) is the complement of \( A \).

We need to find:

1. \( P(A \cap B) \) (the probability of \( A \) and \( B \) occurring together).
2. \( P(A \cup B) \) (the probability of \( A \) or \( B \) occurring).

### Step 1: Find \( P(A) \)

Since \( P(\bar{A}) = 0.40 \), we can use the fact that:

\[
P(A) = 1 - P(\bar{A}) = 1 - 0.40 = 0.60
\]

### Step 2: Find \( P(A \cap B) \)

Since \( A \) and \( B \) are independent events, the probability of their intersection is given by:

\[
P(A \cap B) = P(A) \cdot P(B)
\]

Substitute the values we know:

\[
P(A \cap B) = 0.60 \cdot 0.70 = 0.42
\]

### Step 3: Find \( P(A \cup B) \)

The probability of the union of two events is given by:

\[
P(A \cup B) = P(A) + P(B) - P(A \cap B)
\]

Substitute the known values:

\[
P(A \cup B) = 0.60 + 0.70 - 0.42 = 1.30 - 0.42 = 0.88
\]

### Final Answer:

- \( P(A \cap B) = 0.42 \)
- \( P(A \cup B) = 0.88 \)

---

Would you like more details or clarifications? Here are some related questions:

1. How would the probabilities change if \( A \) and \( B \) were dependent events?
2. What is the significance of the complement \( \bar{A} \) in probability theory?
3. Can you explain why the formula for the union of two events subtracts \( P(A \cap B) \)?
4. How do you calculate conditional probabilities with independent events?
5. What happens if \( P(A) \) or \( P(B) \) exceeds 1, and is that possible?

**Tip:** Independence of events simplifies probability calculations, as their joint probability becomes the product of individual probabilities.

Supongamos que A y B son sucesos independientes tal que P(Ā) = 0.40 y P(B) = 0.70. Hallar P(A ∩ B) y P(A ∪ B).

Learn how to find the probabilities P(A ∩ B) and P(A ∪ B) for independent events with given values of P(Ā) and P(B). This step-by-step guide covers key probability concepts, including complement and multiplication rules, suitable for high school students.

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