# Binomial Distribution for a person walking in straight line

## Homework Statement

Can I measure the probability of a person being at a certain end location after n steps using the binomial distribution if,
probability student goes x=x+3 is 0 <= p <0.5 , x=x-1 is 0<= 0.5 p <1.

## Homework Equations

x=x+3 is 0 <= p <0.5
x=x-1 is 0<= 0.5 p <1

## The Attempt at a Solution

I know that I can do this for x=x+1 0<=p<0.5 and x=x-1 0.5<=p<1
And I know it doesn't work if it's position based, so for example at x=10, the probabilities changed.
But, can I do it if they move a larger distance in one direction than the other such as the example I have posted?
Thanks

Ray Vickson
Science Advisor
Homework Helper
Dearly Missed

## Homework Statement

Can I measure the probability of a person being at a certain end location after n steps using the binomial distribution if,
probability student goes x=x+3 is 0 <= p <0.5 , x=x-1 is 0<= 0.5 p <1.

## Homework Equations

x=x+3 is 0 <= p <0.5
x=x-1 is 0<= 0.5 p <1

## The Attempt at a Solution

I know that I can do this for x=x+1 0<=p<0.5 and x=x-1 0.5<=p<1
And I know it doesn't work if it's position based, so for example at x=10, the probabilities changed.
But, can I do it if they move a larger distance in one direction than the other such as the example I have posted?
Thanks

I think you meant to say that the moves are ##x \to x+3## w.p. ##p## and ##x \to x-1## w.p. ##q = 1-p##; here ##0 < p < 1/2##. (The case ##p=0## should be excluded, because it is trivial---there is no randomness at all.)

After ##n## steps, if the student takes ##k## steps to the right and ##(n-k)## steps to the left, how far to the right has he/she moved? (Of course, a negative distance to the right is a positive distance to the left.)

I did not understand at all the rest of your post, where you talk about x = 10 and larger distances, etc.

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Can I say that probability for any given end point x, the probability of ending up there is :
probability = {factorial(n)/(factorial((n+(x+3))/2)*factorial((n-(x-1))/2))}*{p^(1/2(n+(x+3)) * q^(1/2(n+(x-1)))}
Where p = probability of moving x+3 and q = 1-p

Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
Can I say that probability for any given end point x, the probability of ending up there is :
probability = {factorial(n)/(factorial((n+(x+3))/2)*factorial((n-(x-1))/2))}*{p^(1/2(n+(x+3)) * q^(1/2(n+(x-1)))}
Where p = probability of moving x+3 and q = 1-p

That is not what I get. Just solve for ##k## in terms of ##x## and substitute that into the binomial expression, but do it carefully. (Of course, I assume the starting point is ##x = 0##).

BTW: it is normal in probability to use shorthand notation for binomial coefficients: instead of
$$\frac{a!}{b! (a-b)!}$$
we usually write
$$\binom{a}{b} \;\;\rm{or} \;\; C(a,b) \;\; \rm{or} \;\; {}_aC_b$$
Of course, when we actually want to compute the binomial coefficient, we fall back on the original formula in terms of factorials. However, do what makes you most comfortable; I am just offering advice that you can take or leave.