Question

Suppose you toss a fair coin three times. Which of the following events are independent? Give mathematical justification for your answer.

    A=

{“heads on first toss”}; B=
{“an odd number of heads”}.
A=
{“no tails in the first two tosses”}; B=

    {“no heads in the second and third toss”}.

Answer 1

For two events A and B to be independent

P ( A and B ) = P(A)*P(B)

Sample space of tossing a coin 3 times = {S }= { HHH , HHT , HTH , THH , TTH , THT , HTT , TTT }

Total outcomes = 2*2*2 => 8

#1 A = Heads on first toss = > HHH HHT HTH HTT

P(A) = 4/8 = 1/2

B = Odd no of heads = > HHH HTT THT TTH =

P(B) = 4/8 = 1/2

A and B = Heads on first toss and odd no. of heads = HHH HTT

P( A and B ) = 2/8 = 1/4

1/4 = 1/2*1/2

P( A and B ) = P(A)*P(B) So these events are independent

#2

A = no tails in the first two tosses = HHT HHH

P(A) = 2/8 = 1/4

B = no heads in the second and third toss = { HTT , TTT }

P(B) = 2/8 = 1/4

A and B = no tails in the first two tosses and no heads in the second and third toss => no such events

P(A and B) = 0

P(A and B ) P(A)*P(B)

So A and B are not independent in this case