Question

g.pr.10.3 (mc)

\\(p(a) = 0.25\\), \\(p(b) = 0.6\\), and \\(p(a \cap b) = 0.15\\). are a and b independent events? why or why not?

yes, \\(p(a) + p(b) = p(a \cap b)\\), therefore they are independent events
yes, \\(0.25 \cdot 0.6 = 0.15\\), which is \\(p(a \cap b)\\), therefore they are independent events
no, \\(p(a) + p(b) \
eq p(a \cap b)\\), therefore they are dependent events
no, \\(0.25 \cdot 0.6 = 0.15\\), therefore they are dependent events

Explanation:

Calculate the product of individual probabilities

$$ P(A) \cdot P(B) = 0.25 \cdot 0.6 = 0.15 $$

Compare with the intersection probability

$$ P(A \cap B) = 0.15 $$

$$ P(A) \cdot P(B) = P(A \cap B) $$

Determine independence

$$ \text{Since } P(A \cap B) = P(A) \cdot P(B), \text{ the events } A \text{ and } B \text{ are independent.} $$

Answer:

• Yes, \(P(A) + P(B) = P(A \cap B)\), therefore they are independent events
• Yes, \(0.25 \cdot 0.6 = 0.15\), which is \(P(A \cap B)\), therefore they are independent events (Correct answer)
• No, \(P(A) + P(B)

eq P(A \cap B)\), therefore they are dependent events

• No, \(0.25 \cdot 0.6 = 0.15\), therefore they are dependent events