Let W be a random variable giving the number of heads minus the
number of tails in three tosses of a coin. (i) List the elements of the
sample space S for the three tosses of the coin and to each sample
point assign a value w for W. (ii) Find the probability distribution of
the random variable W assuming that the coin is biased so that a
head is twice as likely to occur as a tail

Question

Let W be a random variable giving the number of heads minus the 
number of tails in three tosses of a coin. (i) List the elements of the 
sample space S for the three tosses of the coin and to each sample 
point assign a value w for W. (ii) Find the probability distribution of 
the random variable W  assuming that the coin is biased so that a 
head is twice as likely to occur as a tail

zarkon · Answer

do you know what $S$ is?

paxpolaris · Answer

(i)
$$\begin{matrix}\text{Sample} & W
\ HHH & 3 
\ HHT & 1
\ HTH & 1
\ HTT & -1
\ THH & 1
\ THT & -1
\ TTH & -1
\ TTT & -3\end{matrix}$$

paxpolaris · Answer

i suppose you were looking for something like that...

paxpolaris · Answer

(ii) hint:
a sequence  with 1H is twice as likely than the sequence with no H (TTT).
a sequence  with 2H is 2*2=4times as likely than the sequence with no H.
the sequence  with 3H is 2*2*2=8times as likely than the sequence with no H.