A coin is tossed three times. Let the events A,B and C be defined as follows:
A=first toss is head, B=second toss is head, and C= exactly two heads are tossed in a row.
Check the independence of (i) A and B
(ii) B and C
(iii) C and A
Sample space S={(HHH),(HHT),(HTH),(HTT),(THH),(THT),(TTH),(TTT)}
A=first toss is head
A={HHH,HTH,HHT,HTT}
B=second toss is head
B={HHH,HHT,THT,THH}
A∩B={HHH,HHT}
Therefore A and B are independent events.
(ii) Sample space S={(HHH),(HHT),(HTH),(HTT),(THH),(THT),(TTH),(TTT)}
B=second toss is head
P(B)={HHH,HHT,THT,THH}
C= exactly two heads are tossed in a row.
P(C)={HHT,THH}
A∩B={HHT,THH}
Therefore B and C are not independent events.
(iii) Sample space S={(HHH),(HHT),(HTH),(HTT),(THH),(THT),(TTH),(TTT)}
C= exactly two heads are tossed in a row.
C={HHT,THH}
A=first toss is head
A={HHH,HTH,HHT,HTT}
A∩B={HHT}
Therefore A and C are independent events.