Let's break down and solve the problems based on the image provided.

Problem 1:

Let $A$ and $B$ be events with $P(A) = 0.6$, $P(B) = 0.9$, and $P(A \cap B) = 0.5$.

(a) Are $A$ and $B$ independent? Explain.
Two events are independent if: $P(A \cap B) = P(A) \cdot P(B)$ Let's check this condition: $P(A) \cdot P(B) = 0.6 \cdot 0.9 = 0.54$ Since $P(A \cap B) = 0.5$ and $0.54 \neq 0.5$, the events $A$ and $B$ are not independent.

(b) Compute $P(A \cup B)$.
We use the formula for the union of two events: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ Substitute the given values: $P(A \cup B) = 0.6 + 0.9 - 0.5 = 1.0$

(c) Are $A$ and $B$ mutually exclusive? Explain.
Two events are mutually exclusive if: $P(A \cap B) = 0$ Since $P(A \cap B) = 0.5 \neq 0$, the events $A$ and $B$ are not mutually exclusive.

Problem 2:

Let $A$ and $B$ be events with $P(A) = 0.4$, $P(B) = 0.5$, and $P(A \cup B) = 0.6$.

(a) Compute $P(A \cap B)$.
We use the same formula for the union of two events: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ Substitute the known values: $0.6 = 0.4 + 0.5 - P(A \cap B)$ $P(A \cap B) = 0.9 - 0.6 = 0.3$

(b) Are $A$ and $B$ mutually exclusive? Explain.
Since $P(A \cap B) = 0.3 \neq 0$, the events $A$ and $B$ are not mutually exclusive.

(c) Are $A$ and $B$ independent? Explain.
Check if $P(A \cap B) = P(A) \cdot P(B)$: $P(A) \cdot P(B) = 0.4 \cdot 0.5 = 0.2$ Since $P(A \cap B) = 0.3 \neq 0.2$, the events $A$ and $B$ are not independent.

Problem 3:

Let $A$ and $B$ be events with $P(A) = 0.5$, $P(B) = 0.3$, and $P(A \cup B) = 0.8$.

(a) Compute $P(A \cap B)$.
Using the formula for the union of two events: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ Substitute the known values: $0.8 = 0.5 + 0.3 - P(A \cap B)$ $P(A \cap B) = 0.8 - 0.8 = 0$

(b) Are $A$ and $B$ mutually exclusive? Explain.
Since $P(A \cap B) = 0$, the events $A$ and $B$ are mutually exclusive.

(c) Are $A$ and $B$ independent? Explain.
For independence, we check if $P(A \cap B) = P(A) \cdot P(B)$. Since $P(A \cap B) = 0$ and $P(A) \cdot P(B) = 0.5 \cdot 0.3 = 0.15$, they are not independent because $0 \neq 0.15$.

Do you want more details or have any questions?

Related Questions:

1. What is the difference between mutually exclusive and independent events?
2. How do you compute $P(A \cup B)$ if $A$ and $B$ are mutually exclusive?
3. Can two events be both mutually exclusive and independent? Why or why not?
4. How would the answers change if $P(A \cap B) = P(A) \cdot P(B)$?
5. What are some real-world examples of independent and mutually exclusive events?

Tip:

Always start by checking if events are mutually exclusive or independent before calculating probabilities, as this affects the formulas you'll use.