Problem: Pairwise and mutual independence

Problem

• If we have three events \(A,B,C\) then we say that they are pairwise independent if \(A\) and \(B\) are independent, \(A\) and \(C\) are independent and \(B\) and \(C\) are independent.
• We say that they are mutually independent if they are pairwise independent and \(P(A∩B∩C)=P(A)P(B)P(C)\) .
• Toss a coin twice with \(S=\) {HH,HT,TH,TT} . Define
• \(A\) is "head in the first toss, \(A=\) {HH,HT}
• \(B\) is "head in the second toss, \(B=\) {HH,TH}
• \(C\) is “b oth tosses give the same outcome ”, \(C={HH,TT}\)
• Show that \(A,B,C\) are pairwise independent but not mutually independent.
• Note: The term “independent events” typically refers to mutually independent events, not pairwise independent events when we have three or more events.