When I swing at a nail, I drive it all the way in with probability 1/2. With probability 1/4, I hit it half-way in, and with 1/4 probability I miss it entirely. Hitting it half-way in twice will drive it all of the way in. I'm pretty sure that if I swing 4 times at a nail, I'll get it all the way in almost all the time. Let's see if I'm right.
How many sequences of 4 swings could leave the nail still not knocked all the way in?