Question

Let A and B be events with P(A) = 0.7, P(B) = 0.3 and A(A and B) = 0.12.
Are A and B independent? explain

Answer 1

compute P(A or B)

Answer 2

Are A and B mutually exclusive? Explain

Answer 3

To determine whether events \( A \) and \( B \) are independent, we need to check if the following condition holds:

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

Where:
- \( P(A) \) is the probability of event \( A \).
- \( P(B) \) is the probability of event \( B \).
- \( P(A \cap B) \) is the probability of both events \( A \) and \( B \) occurring.

Given:
- \( P(A) = 0.7 \)
- \( P(B) = 0.3 \)
- \( P(A \cap B) = 0.12 \)

Now, we compute \( P(A) \cdot P(B) \):

\[
P(A) \cdot P(B) = 0.7 \cdot 0.3 = 0.21
\]

Next, we compare \( P(A \cap B) \) with \( P(A) \cdot P(B) \):

- \( P(A \cap B) = 0.12 \)
- \( P(A) \cdot P(B) = 0.21 \)

Since \( P(A \cap B) = 0.12 \) is not equal to \( P(A) \cdot P(B) = 0.21 \), we conclude that:

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

Thus, events \( A \) and \( B \) are **not independent**.

Answer 4

To compute \( P(A \cup B) \), which is the probability of either event \( A \) or event \( B \) occurring, we can use the following formula:

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

Where:
- \( P(A) \) is the probability of event \( A \).
- \( P(B) \) is the probability of event \( B \).
- \( P(A \cap B) \) is the probability of both events \( A \) and \( B \) occurring.

Given:
- \( P(A) = 0.7 \)
- \( P(B) = 0.3 \)
- \( P(A \cap B) = 0.12 \)

Now, we can substitute these values into the formula:

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

\[
P(A \cup B) = 0.7 + 0.3 - 0.12
\]

Calculating this gives:

\[
P(A \cup B) = 1.0 - 0.12 = 0.88
\]

Therefore, the probability \( P(A \cup B) \) is \( 0.88 \).

Answer 5

To determine if events \( A \) and \( B \) are mutually exclusive, we need to check the definition of mutually exclusive events. Two events are said to be mutually exclusive if they cannot occur at the same time, which mathematically implies:

\[
P(A \cap B) = 0
\]

This means that the probability of both \( A \) and \( B \) happening together is zero.

Given the information:

- \( P(A \cap B) = 0.12 \)

Since \( P(A \cap B) = 0.12 \) is not equal to zero, it indicates that events \( A \) and \( B \) can occur simultaneously with a probability of \( 0.12 \).

**Conclusion:** Events \( A \) and \( B \) are **not mutually exclusive**, as there is a non-zero probability that both events can happen at the same time.