Am I applying the binomial equation correctly?

Question

Let's say we have a process where I move forward with probability $p$ and backwards with probability $(1-p)$.

Then, is the probability that I end up one step forward after three steps $\binom{3}{2}p^2(1-p)^1$ or am I misusing the binomial equation?

Answer 1

You have $8$ options after $3$ steps:

$$BBB,BBF,BFB,FBB,BFF,FBF,FFB,FFF$$

The probability of each of these is easy to calculate, and so, yes, you are right.

Answer 2

Your answer is exactly right.

More generally (just in case you are curious), to end up $k$ steps forward after $n$ steps, $k+n$ has to be even, and we have to go $\frac{n+k}{2}$ steps forward and $\frac{n-k}{2}$ steps back. The probability of this happening is $$ \binom{n}{(\frac{n+k}{2})} p^{\frac{n+k}{2}} (1-p)^{\frac{n-k}{2}}. $$

Plugging in $k = 1$ step forward after $n = 3$ steps, we get $\binom{3}{2} p^2 (1-p)^1$, again confirming your answer is right.