Exercise · ZW-INDEPENDENCE-OF-EVENTS-009 · ★★★ Advanced

Let \(A\) and \(B\) be independent events with positive probabilities. Can they be mutually exclusive?

Choose one option.

Show answer

B. No: \(P(A\cap B)=P(A)P(B)>0\), but exclusive events would give 0

Explanation

Independence gives \(P(A\cap B)=P(A)P(B)>0\); exclusivity requires \(P(A\cap B)=0\). They are different ideas: exclusive events with positive probability are dependent.

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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?