understanding when to use the binomial theorem.. or using it properly in different questions

Question

So I have a problem that says:

A fair coin is tossed repeatedly. $Y$ is the number of the toss on which the 3rd $H$ appears.

find $P(Y=4)$

When I approached this problem I thought I'd try the binomial theorem and used $p = .5$ and $X = 4$ and $n = 3$. but when I looked at the answer i do get $\left(\begin{array}{c}3\\2\end{array}\right)\left(\dfrac{1}{2}\right)^4$, but no $(1-p)^{n-x}$ like I the binomial distribution formula said. The answers also showed the same type of solution for $Y = 5, 6, 7,$ etc.

so why is the $(1-p)^{n-x}$ part taken off?

Answer 1

To get the third head on the fourth toss, one of the following sequences of tosses must occur: HHTH, HTHH, THHH. It's now clear that the probability is $3/16$.

In general, don't forget the $Y$-th toss must be a head.

Answer 2

More generally, when tossing a (not necessarily fair) coin, the $k$'th Heads occurs on the $n$'th toss when there are exactly $k-1$ Heads in the first $n-1$ tosses and then the $n$'th toss is Heads. Thus if tosses are independent, each with probability $p$ of Heads, the probability is ${n-1 \choose k-1} p^k q^{n-1-k}$.