Question

(a) A fair coin is tossed five times. Let E be the event that an odd number of tails occurs, and let F be the event that the first toss is tails. Are E and F independent?

(b) A fair coin is tossed twice. Let E be the event that the first toss is heads, let F be the event that the second toss is tails, and let G be the event that the tosses result in exactly one heads and one tails. Are E,F, and G independent?

Answer 1

P(E)=P(odd number of tails)=P(1 tails)+P(3 tails)+P(5 tails)

==0.5

P(F)=P(first toss is tail)=0.5

P(E n F)=P(first toss is tail and (0 tail in next 4+2 tail in next four +4 tail in next four))

=

=0.5*0.5=0.25

as P(E)*P(F)=P(E n F) ; therefore E and F are independent,

b)

P(E)=P(first toss is heads)=0.5

P(F)=P(second toss is tails)=0.5

P(G)=P(exactly one head and one tails)=P(first head and second tail)+P(2nd head and 1st tail)

=0.5*0.5+0.5*0.5=0.5

P(E n F nG)=P(first head and second tail) =0.5*0.5=0.25

here as P(E n F nG) is not equal to P(E)*P(F)*P(G) ; therefore E,F and F are not independent,