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?

Answer 1

\[P(4\ misses\ out\ of\ 4\ attempts)=(\frac{1}{4})^{4}=\frac{1}{256}\]
\[P(3\ misses\ out\ of\ 4\ attempts)=\left(\begin{matrix}4 \\ 3\end{matrix}\right)(\frac{1}{4})^{3}\times \frac{3}{4}=\frac{3}{64}\]
\[P(3\ misses\ plus\ 1\ halfway)=\frac{3}{64}\times\frac{1}{4}\]
\[P(4\ misses\ or\ halfway\ after\ 4\ attempts)=\frac{1}{256}+\frac{3}{64}\times\frac{1}{4}=you\ can\ calculate\]

Answer 2

Thanks!

Answer 3

You're welcome :)