Independent events

Summary

Definition

• Two events \(A\) and \(B\) are said to be independent if \(P(A∩B)\) \(=P(A)P(B)\)

Example

• We toss a coin twice. \(S=\) {HH,HT,TH,TT} where H means head and T means tail.
• The outcome HT for Example is short for "head in the first toss and tail in the second toss".
• We assume the coin is fair such that each outcome is equally likely.
• \(A\) is the event "head in the first toss, \(A=\) {HH,HT} and \(B\) is the event "head in the second toss, \(B=\) {HH,TH} .
• Then \(A\) and \(B\) are independent events :
• \(A∩B=\) {HH} and \(P(A∩B)=1/4\) .
• \(P(A)=2/4=1/2\) , \(P(B)=1/2\) and \(P(A)P(B)=1/4\) .

Comments

• If \(A\) and \(B\) are independent events then \(P(A|B)=P(A)\) . This means that knowing that \(B\) is true will not affect the probability of \(A\) – the events are unrelated.
• Do not confuse independent events with mutually exclusive events.