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We are given that events \( A \) and \( B \) are independent, with the following probabilities:
\[
P(A) = 0.70 \quad \text{and} \quad P(B) = 0.10.
\]

We are asked to find:

1. \( P(A \cap B) \) (the probability of the intersection of \( A \) and \( B \)).
2. \( P(A \cup B) \) (the probability of the union of \( A \) and \( B \)).

### Step 1: Calculate \( P(A \cap B) \)
For independent events, the probability of their intersection is given by:
\[
P(A \cap B) = P(A) \cdot P(B).
\]
Substituting the given values:
\[
P(A \cap B) = 0.70 \cdot 0.10 = 0.07.
\]

### Step 2: Calculate \( P(A \cup B) \)
The probability of the union of two events is given by the formula:
\[
P(A \cup B) = P(A) + P(B) - P(A \cap B).
\]
Substituting the known values:
\[
P(A \cup B) = 0.70 + 0.10 - 0.07 = 0.80 - 0.07 = 0.73.
\]

### Final Answers:
\[
P(A \cap B) = 0.07,
\]
\[
P(A \cup B) = 0.73.
\]

Would you like further clarification or details?

Here are 5 related questions to deepen your understanding:
1. How do you calculate \( P(A \cap B) \) for dependent events?
2. What is the relationship between mutually exclusive and independent events?
3. How would the formula for \( P(A \cup B) \) change if events \( A \) and \( B \) were mutually exclusive?
4. Can the probability of the union of two events ever exceed 1? Why or why not?
5. What does it mean if \( P(A \cap B) = 0 \)?

Tip: For independent events, knowing the probability of one event happening gives no information about the probability of the other event occurring.

Supongamos que A y B son sucesos independientes tal que P(A) = 0.70 y P(B) = 0.10. Hallar P(A ∩ B) y P(A ∪ B).

Learn how to calculate the intersection P(A ∩ B) and union P(A ∪ B) of two independent events, where P(A) = 0.70 and P(B) = 0.10. Step-by-step explanation for students at the college or university level.

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