Is there a way to work out the probability of a name being picked during the reaping and do the odds change depending on the District you come from or are there certain restrictions on how many people can live in each District?
Unfortunately, I don't think you will be able to get an accurate answer, because of the way the reaping is structured, and the addendums that go with it, plus the unknown quantities.
Here are the salient points for the reaping:
- From age 12 to 18, your formula is 1 + (Age-12).
- If you are poor and receiving subsidies (tesserae), one extra chance for every family member at age 12. So a family of 5 gets another 5 entries
- Multiply the tesserae number by your chances. So at 15, your base chance is 4. Tesserae is 4x5, so the formula is base chances + (tesserae * base)
So, for Katniss, just her base number at 16 years (Age at the 74th Hunger Games drawing) would be 5. She volunteered for tesserae every year for a family of three, so that would be 15. She would have had 20 chances in the bowl, Prim had 1.
However, without knowing the exact proportion (or even a close approximation) of the boy/girl and age census figures, as well as who used tesserae chances, you really can't get much more than a wild approximation.
As far as the Careers that volunteer, technically your chances are the same. You can't say that the odds are zero because you think that someone will volunteer. The odds are not zero until that actually happens, so when they put the names in the bucket in a Career district, the odds are whatever they are.
I did find an interesting PDF (located here) that broke down the statistics of it as a middle school exercise.
The Odds are ever in your favor!
The answer to your question is; No. You can not tell the exact odds withou tthe exact census from Panem. You would need to know the age brackets of everyone to be able to determine the number and gender of the people who are eligible to be chosen for the Reaping.
However, in an overall general sense, assuming everyone in every district is eligible, it would break out somewhat like this:
The odds of being picked during the Reaping depend solely on the population of the District, and each District did have different census numbers. Reading Moshe's answer here, it appears that a "small" district has roughly 8,000 people, and a "large" district has roughly 32,000 people. I've found no breakdown of gender, but US gender is roughly 50/50 so we'll assume the same. That means:
in a small district:
Odds of being chosen: 1 in 4,000 or .025%
In a large district:
Odds of being chosen: 1 in 16,000 or .00625%
While these numbers aren't exact by any means, it's logical to believe that the actual odds are higher than calculated above, since the number of eligible citizens (those who fall within the eligible age bracket, which would be the number in the denominator) would be lower. Much like 1/2 is larger than 1/4, which is larger than 1/8. As the denominator increases, the ratio (or "odds") decreases.