Problem source

A pack of cards consists of  $n+1$ cards, 
which are printed with the integers from 
$0$ to $n$.   A~game consists of drawing cards  repeatedly at 
random from the pack until the 
card printed with 0 is drawn, at which point  the game ends. 
After each draw, the player 
receives $\pounds 1$ if the card  drawn shows any of the 
integers from $1$ to $w$ inclusive but receives nothing 
if  the card  drawn shows any of the 
integers from $w+1$ to $n$ inclusive.
\begin{questionparts}
\item[\bf (i)]  In one version of the game, each card drawn is replaced immediately
and randomly in the pack.
Explain clearly why the probability that the player 
wins a total of exactly $\pounds 3$ 
is equal to the probability of the following event 
occurring:
out of the first four cards drawn which show 
 numbers in the range $0$ to $w$, 
the numbers on the first three are non-zero and the  
number on the fourth is zero.
Hence show that the probability that the player 
wins a total  of exactly  $\pounds 3$ is equal to $\displaystyle \frac{w^3}{(w+1)^4}$. 
 
Write down the probability that the player wins a total of exactly 
$\pounds r$ and hence find the  expected total win.  
 
\item[\bf (ii)] In another version of the game,  
each card drawn is removed from the pack.
Show that the  expected total win in this version is 
half of the expected total win in the other version. 
\end{questionparts}