# Ant problem with discrete combinatorical background.

an ant can move along a grid in $\mathbb{Z}^2$. But the ant can only go upwards and to the right(with equal probability). The ant starts in the point $(0,0)$, but there is an electrical wire from $(0,14)$ to $(23,14)$ and from $(25,0)$ to $(25,12)$ that would kill the ant. The question is: What is the probability that the ant survives?

My problem is that I do not know how to tackle this problem. It appears to me that it is not that easy to find an equation that says how many different paths go to which point, which would certainly simplify this problem, but anyway: I did not come up with something useful so far. Notice, that if the ant wants to survive it must certainly pass the point$(24,13)$.

Is there anybody who has an idea?

As you pointed out, since the ant gets zapped if it reaches $y=14$ with $x<24$, or reaches $x=25$ with $y<13$, there's only one place it can safely be after $37$ steps: $(24, 13)$. Moreover, if it reaches that location, it's safe for all subsequent steps. So, there are $2^{37}$ equally probable ways to choose the first $37$ steps, of which exactly ${{37}\choose{13}}$ are safe. The probability of survival can be calculated from that information (it's about $2.6\%$).