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. 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.

What is the probability I fail to get the nail driven completely in?

Question

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. 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.

What is the probability I fail to get the nail driven completely in?

anonymous · Answer

5/256. Just use casework. This question is relatively easy.

This when all 4 swings are misses (1/4)^4=1/256
3 swings are misses, 1 is halfway (1/4)^3*(1/4)*(4C3)=4/256

1/256+4/256=5/256.
The answer is (5/256)