Question

1. based on these probabilities, are the events independent? explain your reasoning.

Explanation:

To determine if two events \( A \) and \( B \) are independent, we use the definition of independent events: two events are independent if and only if \( P(A \cap B) = P(A) \times P(B) \).

Step 1: Recall the formula for independence

For events \( A \) and \( B \), independence requires:
\[ P(A \cap B) = P(A) \times P(B) \]

Step 2: Apply the formula to the given probabilities

Let’s denote the two events as \( A \) and \( B \). Suppose we know \( P(A) \), \( P(B) \), and \( P(A \cap B) \) (from the problem’s context, though not fully shown here).

• Calculate \( P(A) \times P(B) \).
• Compare this product to \( P(A \cap B) \).

Step 3: Interpret the result

• If \( P(A \cap B) = P(A) \times P(B) \), the events are independent.
• If \( P(A \cap B)

eq P(A) \times P(B) \), the events are dependent.

To fully solve, we need the specific probabilities of \( A \), \( B \), and \( A \cap B \). For example, if \( P(A) = 0.5 \), \( P(B) = 0.4 \), and \( P(A \cap B) = 0.2 \):

• \( P(A) \times P(B) = 0.5 \times 0.4 = 0.2 \), which equals \( P(A \cap B) \), so the events are independent.

If \( P(A \cap B) \) differed from \( P(A) \times P(B) \), they would be dependent.

(Note: Since the original problem’s probabilities are not fully visible, substitute the actual values for \( P(A) \), \( P(B) \), and \( P(A \cap B) \) into the formula to determine independence.)

Answer:

To determine if two events \( A \) and \( B \) are independent, we use the definition of independent events: two events are independent if and only if \( P(A \cap B) = P(A) \times P(B) \).

Step 1: Recall the formula for independence

For events \( A \) and \( B \), independence requires:
\[ P(A \cap B) = P(A) \times P(B) \]

Step 2: Apply the formula to the given probabilities

Let’s denote the two events as \( A \) and \( B \). Suppose we know \( P(A) \), \( P(B) \), and \( P(A \cap B) \) (from the problem’s context, though not fully shown here).

• Calculate \( P(A) \times P(B) \).
• Compare this product to \( P(A \cap B) \).

Step 3: Interpret the result

• If \( P(A \cap B) = P(A) \times P(B) \), the events are independent.
• If \( P(A \cap B)

eq P(A) \times P(B) \), the events are dependent.

To fully solve, we need the specific probabilities of \( A \), \( B \), and \( A \cap B \). For example, if \( P(A) = 0.5 \), \( P(B) = 0.4 \), and \( P(A \cap B) = 0.2 \):

• \( P(A) \times P(B) = 0.5 \times 0.4 = 0.2 \), which equals \( P(A \cap B) \), so the events are independent.

If \( P(A \cap B) \) differed from \( P(A) \times P(B) \), they would be dependent.

(Note: Since the original problem’s probabilities are not fully visible, substitute the actual values for \( P(A) \), \( P(B) \), and \( P(A \cap B) \) into the formula to determine independence.)