As you say, the problem here is mainly about protecting the confidntiality of your idea when you pitch it to a betting company. And the problem with copyright is that it is not idea itself which is protected, but the actual fixation of the idea in some manner. Thus if you write it down, what you write - the selection and order of the words you use - is what is protected from un-authorised copying, rather than the idea behind them. And you can't apply for a patent on a business process, which is what your idea seems to resemble.
So I think the best way to approach this is use the common law concept of confidentiality. This is often used when pitching an idea such as a game show concept to a TV company.
To do this properly, I suggest you need to speak to a solicitor who specialises in media, commercial and IP law who can help you draw up a document which when signed by someone, binds them to keep confidential what they are about to be told. You can get some more information here: http://www.gillhams.com/dictionary/24.cfm. If the company you approach likes the idea you can then licence them to use it while retaining ownership of the idea for further exploitation. However, once one company adopts your idea, and it becomes well-known, there is little you can do to prevent others from copying the idea without paying you, and the initial confidentiality will no longer apply.
I'm afraid that's not an untypical cost of an agreement of this type. With that you would be getting a high degree of confidence that the agreement would be tailored to your needs and not have any loopholes. It should also be covered by the lawyer's professional insurance.
A less foolproof but cheaper alternative would be to use what is known as a boilerplate agreement which you tailor yourself. An example can be found here: http://www.own-it.org/contracts/3