# Binomial Distribution for a person walking in straight line

• CourtneyS
In summary, the conversation discusses the use of the binomial distribution to measure the probability of a person being at a certain end location after a given number of steps, with the person having a probability of moving either x+3 or x-1 at each step. The poster suggests using a formula involving factorial expressions and probabilities p and q for this calculation, but it is advised to use a more careful approach to arrive at the correct solution.

## 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

CourtneyS said:

## 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.

Last edited:
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

CourtneyS said:
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.

## 1. What is the binomial distribution for a person walking in a straight line?

The binomial distribution for a person walking in a straight line is a probability distribution that describes the likelihood of a person taking a specific number of steps in a straight line, given a certain number of trials or attempts.

## 2. What factors affect the binomial distribution for a person walking in a straight line?

The factors that affect the binomial distribution for a person walking in a straight line include the length of the steps taken, the number of trials or attempts, and the direction of the steps (e.g. left or right).

## 3. How is the binomial distribution for a person walking in a straight line calculated?

The binomial distribution for a person walking in a straight line is calculated using the formula P(x) = (nCx)(p^x)(q^(n-x)), where n is the number of trials, x is the number of successes, p is the probability of success, and q is the probability of failure.

## 4. Can the binomial distribution be used to predict the exact number of steps a person will take in a straight line?

No, the binomial distribution cannot be used to predict the exact number of steps a person will take in a straight line. It provides a probability distribution that shows the likelihood of taking a certain number of steps, but it cannot predict the exact outcome.

## 5. How is the binomial distribution for a person walking in a straight line useful in real life?

The binomial distribution for a person walking in a straight line can be useful in real life for analyzing and predicting outcomes in various scenarios such as sports, games, and even in research studies. It can also be used to understand and improve daily activities that involve walking, such as hiking or navigating through a crowded city.