I've never watched or read The Hunger Games, but I have studied statistics. I performed a Bayesian analysis to determine what beliefs about the situation are mathematically consistent. I was surprised to find that it seems very plausible that the selection process was in fact rigged. However, this depends heavily on the assumptions one makes about the situation. Here is my full analysis:

Let S denote the event "Prim is selected" and R denote the event "the selection process is rigged to select Prim."

We know that Prim was in fact selected and we want to know whether the selection was rigged. We can't know for certain, so instead we'll ask, "how likely is it that the selection process was rigged if Prim was selected?" This corresponds to the symbol P(R | S), pronounced "probability of R given S."

Now, we use a formula called Bayes' theorem to express this probability in terms of other variables:

P(R | S) = P(S | R) * P(R) / P(S)

The formula expands further using the Total Law of Probability:

P(R | S) = P(S | R) * P(R) / (P(S|R) * P(R) + P(S | -R) * P(-R))

The symbols -S and -R stand for the events in which S and R respectively do not happen.

Let us assume that if the selection is rigged, Prim has a 100% chance of being selected. This corresponds to the statement P(S | R) = 1. Additionally, let us assume that if the selection is fair, then Prim has a 1 in 5,000 chance of being selected. This corresponds to the statement P(S | -R) = 1/5000 = 0.0002. Let's simplify:

P(R | S) = 1 * P(R) / (1 * P(R) + 0.0002 * P(-R))

P(-R) can be substituted with 1-P(R), leaving us with only one variable on the right side: P(R).

P(R | S) = P(R) / (P(R) + 0.0002 * (1-P(R))

We can simplify a bit further:

P(R | S) = P(R) / (0.9998 * P(R) + 0.0002)

Now, we need to address exactly what this P(R) refers to. It is what is called a prior probability. What it means is a little complicated to explain, but the simplest way to think of it is what we believe about an event before we collect any data on it. In this case, P(R) is what our our estimate of probability that the selection is rigged to select Prim would be, if we didn't actually know the outcome.

In other words, imagine you get to the page where they're announcing the tributes, and you stop reading just before you see Prim's name. You stop and think to yourself, for no particular reason, "hmm, I wonder if the selection was rigged for Prim to become tribute. I would say that there's an X% chance of that. Let's keep reading and find out." X is our prior for R.

Obviously, there is no one correct answer for this value. Indeed, finding a suitable prior is one of the challenging aspects of Bayesian statistics. Like I said at the beginning, I can't define an authoritative answer for the value of P(R | S), much less state with certainty whether R happened or not. What I can do is show how different values of P(R) affect our confidence that R happened. By plugging in difference values of P(R) into the equation above, we can see exactly how our backgrounds beliefs regarding the selection process affect our conclusions once we know the outcome.

For example, let's say that right before seeing Prim's name in the tribute selection scene, we believe that there is a 1 in 10 chance that the selection process was rigged to select Prim, i.e., P(R) = 0.1. Plugging this into our equation yields:

P(S | R) = 0.1 / (0.9998 * 0.1 + 0.0002)
P(S | R) = 0.9982

...wow. Even with just a 10% chance of the selection process being rigged, the extremely low likelihood of Prim being selected by random chance leads up to conclude that the selection was, in fact, almost certainly rigged.

Dropping down to a 1 in 100 prior for R, the probability of the selection being rigged is still over 98%. Here's a table showing how our choice of prior affects the posterior (our updated belief, P(S | R)):

P(R)    | P(S | R)
--------+-------
0.1     | 0.9982
0.01    | 0.9805
0.001   | 0.8335
0.0001  | 0.3336
0.00001 | 0.0476

At around 1 in 10,000, it seems fair to call it "unlikely" that the selection process was rigged. At 1 in 100,000, it's not believable.

So, in conclusion: should you believe that the selection was rigged to select Prim? It depends on how probable you think that is to be the case, if you disregard the fact that Prim was selected as evidence. If you believe that the prior probability is higher than 1 in 1,000, then you should also believe that the selection was, in fact, rigged.