Your question could be understood in two ways, depending on your application:

a) you need a real probability [0..1] because you want to use it in a probabilistic model.

in this case, its gonna be really hard to find something truly probabilistic, since your input might be not probabilistic to start with. One way is to use a language modeling approach (e.g. n-gram models, topic models), approximate the underlying distribution of the document, and compute the probability that the query was generated from the document distribution. This is what's at the core of BM25.

b) you just need some normalized score in the range [0,1]. For this, you could say the following: if the query is the exact same as the document, probability should be 1, otherwise it should be less. Thus, use whatever similarity scoring mechanism you like, compute the self-similarity of the document (e.g. 450) and use this for normalizing, i.e. divide the (un-normalized) query-document score (e.g. 32) with this self-similarity, resulting in 32/450. If you have negative scores in your similarity, you could either set them to 0 or transform the scale so that everything is in the positive range. I'd recommend setting to 0, since we're talking sparse features (i.e. words) here.

Hope this helps!

If your query is a document that is compared to another document, then have a look in to Inter-textual distance. It is expressed in [0;1]. However, if the query is much shorter than compared document, then  you might not get what initially intended.

Have a look in to concepts of  Levenstain or Jaccard distance functions. It would help you to cope with some issues you working on.

Good luck

This is further to the comments of Chris and Linas... when the boundaries are lying between 0 and 1, then why not try using the concept of fuzzy sets .... the computation would be more accurate and easy.