If $P(A)=0.60$ and $P(B)=0.20$, then $A$ and $B$ are independent events if
A. $P(A \text{ or } B)=0.12$
B. $P(A \text{ or } B)=0.80$
C. $P(A \text{ and } B)=0.12$
D. $P(A \text{ and } B)=0$
Answer (1)
Two events A and B are independent if P ( A c a pB ) = P ( A ) P ( B ) .
Given P ( A ) = 0.60 and P ( B ) = 0.20 , calculate P ( A ) P ( B ) = 0.60 × 0.20 = 0.12 .
Check each option to see which satisfies the independence condition.
Option C, P ( A and B ) = 0.12 , satisfies the condition. Therefore, the answer is P ( A and B ) = 0.12 .
Explanation
Understand the problem and provided data We are given the probabilities of two events, P ( A ) = 0.60 and P ( B ) = 0.20 . We need to determine the condition for independence of events A and B from the given options.
Recall the condition for independence Recall that two events A and B are independent if and only if P ( A c a pB ) = P ( A ) P ( B ) . Let's calculate P ( A ) P ( B ) .
Calculate P(A)P(B) We have P ( A ) P ( B ) = 0.60 im es 0.20 = 0.12 . Now we will check which of the given options satisfies the condition for independence.
Analyze Option A Option A states P ( A or B ) = 0.12 . This is equivalent to P ( A c u pB ) = 0.12 . We know that P ( A c u pB ) = P ( A ) + P ( B ) − P ( A c a pB ) . If P ( A c u pB ) = 0.12 , then 0.12 = 0.60 + 0.20 − P ( A c a pB ) , which implies P ( A c a pB ) = 0.60 + 0.20 − 0.12 = 0.68 . Since P ( A ) P ( B ) = 0.12 = 0.68 , this option does not satisfy the independence condition.
Analyze Option B Option B states P ( A or B ) = 0.80 . This is equivalent to P ( A c u pB ) = 0.80 . If P ( A c u pB ) = 0.80 , then 0.80 = 0.60 + 0.20 − P ( A c a pB ) , which implies P ( A c a pB ) = 0.60 + 0.20 − 0.80 = 0 . Since P ( A ) P ( B ) = 0.12 = 0 , this option does not satisfy the independence condition.
Analyze Option C Option C states P ( A and B ) = 0.12 . This is equivalent to P ( A c a pB ) = 0.12 . Since P ( A ) P ( B ) = 0.12 , this option satisfies the condition for independence.
Analyze Option D Option D states P ( A and B ) = 0 . This is equivalent to P ( A c a pB ) = 0 . Since P ( A ) P ( B ) = 0.12 = 0 , this option does not satisfy the independence condition.
Conclusion Therefore, events A and B are independent if P ( A and B ) = 0.12 .
Examples
Understanding independent events is crucial in many real-world scenarios. For example, consider a marketing campaign where the success of an online ad ( A ) and a TV commercial ( B ) are being evaluated. If A and B are independent, the impact of one doesn't affect the other, allowing for simpler analysis. Knowing that P ( A ) = 0.60 and P ( B ) = 0.20 , independence is confirmed if P ( A and B ) = 0.12 , meaning the combined success rate is what you'd expect if they truly don't influence each other. This helps in resource allocation and strategy refinement.
