Question

A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed twice, find the probability distribution of number of tails.

Answer 1

It is given that a coin is biased in such a way that heads is 3 times as likely to occur as tails.

Let, X be the probability of getting a tail in the biased coin. Then, P( T )=X.

As it is given that the coin is biased so that heads is 3 times as likely to occur as tails. So, P( H )=3X.

For a biased coin,

P( T )+P( H )=1 X+3X=1 4X=1 X= 1 4

Thus, P( T )= 1 4 and P( H )= 3 4 .

The sample space for the tossing of a coin two times is { HH,TT,HT,TH }.

Let, the number of tails be represented by the random variable X.

P( X=0 ) is the probability of no tail.

P( X=0 )=P( H )×P( H ) = 3 4 × 3 4 = 9 16

P( X=1 ) is the probability of one tail.

P( X=1 )=P( HT )+P( TH ) = 3 4 × 1 4 + 1 4 × 3 4 = 3 16 + 3 16 = 3 8

P( X=2 ) is the probability of two tails.

P( X=2 )=P( TT ) = 1 4 × 1 4 = 1 16

Thus, the probability distribution is as follows:

X	0	1	2
P( X )	9 16	3 8	1 16