We develop expressions for the probability distribution of end-end distances of a polymer chain. We start from a simple random walk in one dimension, and generalize the result to three dimensions.

In one dimension, the number of ways of arriving a distance x from the origin after N steps of unit size (n_{+} in the
positive direction and n_{-} in the negative) is given by a combinatorial expression.

| (1) |

where x = n_{+} - n_{-}, N = n_{+} + n_{-} and so n_{+} = (N + x)∕2, and n_{-} = (N - x)∕2. The probability of this
occurrence is just the number of ways of realizing the occurrence divided by the total number of possible
trajectories.

| (2) |

We consider the natural logarithm of the above probability. Using Stirling’s approximation, N!(N∕e)^{N},
and assuming that x ≪ N we arrive at the probability distribution function, which is the probability p(N,x)dx that
the trajectory of the random walk is terminated within an interval dx from x.

In three dimensions, the probability P(N,R) is just the product of P(N,R_{x})P(N,R_{y})P(N,R_{z})dxdy dz and since
< R^{2} >= Nb^{2},

| (4) |

so the probability of finding the chain end (terminus of an ideal random walk) in a spherical shell between R and
R + dR is P(N,R)4πR^{2} dR.