If I am given the count of users participating in a forum thread on a continuous number of days, how can I can find the probability that a number of users will participate in the same thread on another given day in the future?
The OP is asking how to calculate the probability of someone participating another day in the future. A simple possible solution here is to use a Markov Chain. Define the daily transition matrix to be the probability that a person is active on a forum on day $t$ given they were active on day $t-1$. The posting mentions there is data on users active on threads. Count the number of active people on the thread in successive days. These probabilities are what you need. You matrix is a $2\times2$ matrix where you go from either active on day $t$ to one of either active or inactive on day $t+1$. To make a prediction you can use the most recent set of transitions to predict for a person, or you could assume stationarity and use the steady state matrix of the Markov Chain.
If you have additional information about the users, say a covariate or two you could try modeling the observed transitions as binary variables (whether the person moved states or not) an estimate the probabilities using logistic regression, put those into a Markov Chain and do a similar analysis as the case with no covariates I mentioned above. The case here with covariates is probably easiest with only 1 or 2 dichotomous covariates.