Exercise · ZW-INDEPENDENCE-OF-EVENTS-010 · ★★★ Advanced
If \(A\) and \(B\) are independent, which statement is always true?
Choose one option.
Show answer
- C. \(P(A\cap B^c)=P(A)P(B^c)\)
Explanation
\(P(A\cap B^c)=P(A)-P(A)P(B)=P(A)(1-P(B))=P(A)P(B^c)\): the complement of an independent event is also independent.
Review the concept: Independence of events
Two events \(A\) and \(B\) are independent when knowing that one occurred does not change the probability \(P\) of the other.
\[P(A\cap B)=P(A)\,P(B)\]
More practice problems: independence of events
- ZW-INDEPENDENCE-OF-EVENTS-001 If A and B are independent with P(A)=0.3 and P(B)=0.5, what is P(A∩ B)?
- ZW-INDEPENDENCE-OF-EVENTS-002 Two fair coins are tossed. What is the probability of getting two heads?
- ZW-INDEPENDENCE-OF-EVENTS-003 If P(A)=0.4, P(B)=0.5 and P(A∩ B)=0.2, are they independent?
- ZW-INDEPENDENCE-OF-EVENTS-004 If P(A)=0.5, P(B)=0.4 and P(A∩ B)=0.3, which statement is correct?
- ZW-INDEPENDENCE-OF-EVENTS-005 If A and B are independent with P(A)=0.6 and P(B)=0.5, what is P(A| B)?
- ZW-INDEPENDENCE-OF-EVENTS-006 If A and B are independent with P(A)=0.3 and P(B)=0.2, what is P(A∪ B)?
- ZW-INDEPENDENCE-OF-EVENTS-007 A fair coin is tossed three times. What is the probability of at least one head?
- ZW-INDEPENDENCE-OF-EVENTS-008 A fair die is rolled. Let A: «even» and B: «1 or 2». Are they independent?