Probability of Population A population consists of $50$% men and $50$%  women of a population of $50$ people. A simple random sample (draws at random without replacement) of $4$ people is chosen. Find the chance that in the sample  
i) the fourth person is a woman?
ii) The third person is a woman, given that the first person and fourth person are both men?
My attempt:
i) $P(4\text{th Women})=\frac{\dbinom{25}4}{\dbinom{50}4}=\;$?
ii) Conditionality $P(3\text{rd person Women}\mid\text{The }1\text{st and }4\text{th are both men})$?? Way Forward?

We can split it into, the probability that the first three were all women, $2$ women, $1$ woman, or no women. So
$$P(4\text{th Woman})=\frac{\binom{25}{3}}{\binom{50}{3}}\frac{22}{47}+\binom{3}{1}\frac{\binom{25}{2}\binom{25}{1}}{\binom{50}{2}\binom{48}{1}}\frac{23}{47}+\binom{3}{2}\frac{\binom{25}{1}\binom{25}{2}}{\binom{50}{1}\binom{49}{2}}\frac{24}{47}+\frac{\binom{25}{3}}{\binom{50}{3}}\frac{25}{47}$$
For the second part:
$$\begin{align*}
P(3\text{rd Woman} | \text{$1$st and $4$th Men})&=\frac{P(\text{$3$rd Woman and $1$st and $4$th Men})}{P(\text{$1$st and $4$th Men})}\\
&=\frac{25\cdot24\cdot25\cdot23+25\cdot25\cdot24\cdot24}{25\cdot24\cdot23\cdot22+\binom{2}{1}25\cdot24\cdot25\cdot23+25\cdot25\cdot24\cdot24}
\end{align*}$$
