Consider a two dimensional coordinate system.
A bug is located at point (0,0). Every Minute it is moving either exactly one point up or exactly one point to the right with equal probability.
What is the probability for the bug to be at point (7,5) after 12 Minutes?
My thoughts:
This situation can be modeled with a coin tossed 12 times. The outcomes are heads (h) or tails (t).
So we have $\Omega=\{(r_1,...,r_{12})\ |\ r_i\in\{h,t\}\ \forall\ i\in\{1,...,12\}\}$
with $p(\omega)=\frac{1}{2^{12}}$
Here's where I'm stuck though. Any help would be appreciated!
Hint: Any specific sequence of coin tosses has probability $\frac{1}{2^{12}}$ of happening. How many different sequences lets you end up at the point $(7, 5)$?