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. 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?

Answer 1

Let all such sequences be denoted by a string of 4 numbers: 0, 1, and 2.

0 indicates that you miss the nail.
1 indicates a half-hit.
2 indicates a perfect hit.

In four hits, the nail won't be knocked in completely if the sequence contains any 2s, or if it contains more than one 1.

So for example, 0000 means you miss the nail four times, and 0110 means you hit it in halfway on the second and third swings (and so the nail is completely in by the fourth hit).

The number of sequences is then all the sequences with either four 0s, or three 0s and one 1. How many of these sequences are there?

Answer 2

I need this problem solved urgently and i just dont get it. Please answer the whole thing.

Answer 3

does this mean there are 10 sequences