If you toss a fair coin 5 times and if A there are more TAIL

If you toss a fair coin 5 times, and if A= {there are more TAILS than HEADS}, B =

{there is at least one TAIL}, C = {the 1st toss is a HEAD) and D = {the last toss is a

TAIL}, how many pairs of independent events are there? List the independent pairs.

Given

toss a fair coin 5 times, and if A= {there are more TAILS than HEADS}, B =

{there is at least one TAIL}, C = {the 1st toss is a HEAD) and D = {the last toss is a

TAIL}

first we find the sets

then

A={TTTTT,HTTTT,THTTT,TTHTT,TTTHT,TTTTH,HHTTT,HTHTT,HTTHT,HTTTH,THHTT,THTHT,THTTH,TTHHT,TTHTH,TTTHH}

B={TTTTT,HTTTT,THTTT,TTHTT,TTTHT,TTTTH,HHTTT,HTHTT,HTTHT,HTTTH,THHTT,THTHT,THTTH,TTHHT,TTHTH,TTTHH

,TTHHH,THTHH,THHTH,THHHT,HTTHH,HTHTH,HTHHT,HHTTH,HHTHT,HHHTT,THHHH,HTHHH,HHTHH,HHHTH,HHHHT}

C={HHHHH,,HHTTT,HTHTT,HTTHT,HTTTH,HTTTT,HTTHH,HTHTH,HTHHT,HHTTH,HHTHT,HHHTT,HTHHH,HHTHH

,HHHTH,HHHHT}

D={TTTTT,HTTTT,THTTT,TTHTT,TTTHT,HHTTT,HTHTT,HTTHT,THHTT,THTHT,TTHHT,THHHT,HTHHT,HHTHT,HHHTT,HHHHT}

A intersection B={TTTTT,HTTTT,THTTT,TTHTT,TTTHT,TTTTH,HHTTT,HTHTT,HTTHT,HTTTH,THHTT,THTHT,
THTTH,TTHHT,TTHTH,TTTHH}

A intersection C={HTTTT,HHTTT,HTHTT,HTTHT,HTTTH}

A intersection D={TTTTT,HTTTT,THTTT,TTHTT,TTTHT,HHTTT,HTHTT,HTTHT,THHTT,THTHT,TTHHT}

B intersectionC={HHTTT,HTHTT,HTTHT,HTTTH,HTTTT,HTTHH,HTHTH,HTHHT,HHTTH,HHTHT,HHHTT,HTHHH,HHTHH
,HHHTH,HHHHT}

B intersectionD={TTTTT,HTTTT,THTTT,TTHTT,TTTHT,HHTTT,HTHTT,HTTHT,THHTT,THTHT,TTHHT,THHHT,HTHHT,HHTHT,HHHTT,HHHHT}

C intersection D={HHTTT,HTHTT,HTTHT,HHTHT,HHHTT.,HTTTT,HTHHT,HHHHT}

since

P(A)=16/32 =0.5

P(B)=31/32

P(C)=16/32 =0.5

P(D)=16/32=0.5

P(A intersection B)=16/32=0.5

P(A intersection C)=5/32

P(A intersection D)=11/32

P(B intersection C)=15/32

P(B intersection D)=16/32 =0.5

P(C intersection D)=8/32=0.5

P(A)*P(B)=0.5*31/32=15.5/32

P(C)*P(A)=0.5*0.5=0.25

P(D)*P(A)=0.5*0.5=0.25

P(B)*P(C)=0.5*31/32 =15.5/32

P(B)*P(D)=0.5*31/32=15.5/32

P(C)*P(D)=0.5*0.5=0.25

since

P(C intersection D)=P(C)*P(D)=0.25

hence C and D are independent