Question

Can someone please help me with this tricky question in statistics??

stuck on it for quite long, please help if you can:

A drunk man is walking in random steps along an axis with the points +-1, +-2,+-3,... each step he does is in the length of 1 unit with the probability of 0.4 forward and 0.6 backwards(the steps are undependable). X will mark his placement on the axis after 50 steps.

1) what is p{x=-10}?

2)What are the odds that his last step(50th) will be at -27 ?

3)Assuming the the chances of the drunken man of falling in each step is 0.01: if the drunken man walks for 2000 steps, what are the odds in estimation that he will fall exactly 23 times?

Answer 1

See below:

Explanation:

This sets up as a binomial probability. When working with this type of thing, I like to start with this relation:

`sum_(k=0)^(n)C_(n,k)(p)^k(1-p)^(n-k)=1`

So we have 50 steps that the drunk is taking, giving `n=50`. We also have that a step forward has probability of `p=.4`. All we need now is `k`.

1

It turns out that we can find `k` by looking at where the man ends up. If over the course of 50 steps he ends up at `x=-10`, we know he's taken 30 steps backwards and 20 steps forwards - which gives `k=20`. We don't care about the series of steps he took (whether FBFBBBFBFF... or whatever) because the combination term of the binomial will find all the possible ways of making those steps.

Therefore we have:

`C_(50,20)(.4)^20(.6)^(30)~~0.1146`

2

We're now looking at `x=-27`

To get to `-27`, our drunk needs to have taken some number of steps to get there. For instance, let's say our drunk for the first 48 steps goes back and forth between 1 and 0. Now he has 2 steps left to take. Can he end up on `x=1`? No.

In fact, because he is taking an even number of steps, it is impossible to land on the 50th step on a value of `x` that is odd.

The probability of landing on `x=-27` on the 50th step is 0.

3

Here we have `n=2000, p=0.01`, and we're looking at `k=23`:

`C_(2000,23)(.01)^23(.99)^(1977)`

I typed this into google calculator and it came up with an error! Google spreadsheet, however, handled it fine. It gives `~~0.0671`