How to estimate probability of a bill passing the U.S House of Representatives For a bill to pass the U.S. House of Representatives, a simple majority of members have to vote yes. If every member votes this will be 218 of 435. If some members vote "present", their votes are not tallied, and a majority of the remaining votes need to be yes.
Say I have a probability assigned to the likelihood of every individual member voting for, abstaining, and voting against a bal (e.g. $P(\text{member A will vote yes})=.50$, $P(\text{member A will vote no})=.50$ and so on). How do I then calculate the probability (through direct calculation or through simulation) of the bill passing? 
 A: If we assume independence of Reps' votes, then the problem is easy: each Rep is represented as a Bernoulli trail with a specific probability of voting "yes." You can estimate a probability of passage by repeating many such Bernoulli trials.
Voting "yes" is modeled here, because you've asked about a bill's passage, as opposed to the complete vote tally.
The complement of a "yes" vote is voting "not yes," which encompasses No and Present votes: by the model you've outlined, $P(yes)+P(no)+P(present)=1$. Notably, if you go this route, you've ignored the possibility that a rep is absent (this can happen for any number of reasons: death, travel, or taking a different post like Rep. Price when he became HHS secretary -- his House seat will be empty until the special election, so there are effectively 434 seats as of this writing).
If you want to simulate each option, you'll need to use a multinomial probability distribution instead but the mechanics are precisely the same.

This R code should get you started.
set.seed(13)

B <- 2000 # number of simulations to run
p <- rbeta(435, 2, 2) # probability a rep votes yes
expr <- expression(rbinom(435, size=1, prob=p)) # expression object containing number of trials

results <- replicate(B, eval(expr)) # simulation results

num_votes <- colSums(results)
sum(num_votes > 435/2)/B # estimated prob that it passes

png('passage.png')
    plot(density(num_votes, from=0, to=435), xlab="Yes Votes", main="Histogram of Simulation results")
    abline(v=435/2)
dev.off()

This model uses a beta distribution over vote probabilities, which means that the probability of voting yes is actually a beta-binomial distribution, by the property of conjugacy. This observation could streamline the code, but I left it this way for clarity. Changing p to match the real data is just a drop-in fix this way.
If votes are assumed to be dependent in some way, things are more complicated. You'll need a model that accounts how each member's vote is influenced by their peers. This is more realistic because Reps' votes aren't independent:


*

*They tend to vote in blocs (Freedom Caucus, Tuesday Group, Black Caucus, etc.).

*They "horse trade," 

*They face pressure from leadership, the national party, their state's leadership, and the president to vote a particular way. 

*And a bill typically won't be put to a vote unless House leadership can guarantee passage (see also the Hastert Rule), so getting a whip count right is important to avoiding embarrassment -- a model of random voting throws that into disarray. 

*To the extent that Reps vote their constituency, and different constituencies share values, the feelings "back home" can make a big difference, and undermine the independence assumption. Rural Ohio and rural Pennsylvania probably have more in common than rural Ohio and Northern Virginia, so Reps' votes from rural Ohio are a reasonable proxy for Reps' votes in Pennsylvania but not Northern Virginia.

*Even whip counts can influence the margin of a vote: sometimes leadership would prefer for a measure to fail by 10 or more votes instead of it failing by 1, since the dealmaking to get those other 9 votes has costs and tradeoffs which will all be for naught if the measure fails anyway. (Whether a doomed bill makes it to the floor at all depends on the aims of the leadership: the GOP staged numerous symbolic ACA repeal votes, knowing full well that the Senate and/or President would kill the measure, while in other contexts, doomed measures are remanded to committees for revisions.)

*Additionally, the actions of the Senate and President can mean that stunt votes like the 2014 ACA repeal attempts can be undertaken while the members know full well that it will never become law -- so the contents and effects of the bill hardly matter at all. This implies that the Senate's and President's predicted actions will directly bear on how these votes happen: Reps can vote in support of a measure they know to be flawed or incomplete to please the leadership and never face any policy blowback because it will never become law. On the other hand, if the Senate and President are likely to approve the measure, then its consequences and contents are important. This is essentially why ACHA passage is suddenly so fraught after being a cornerstone of the GOP platform for years.
Accounting these factors will make the problem more challenging; however, ignoring dependence, especially in this example, where there are many well-known causes of dependence, will result in a model that is extremely misleading.
A: This is a fantastically difficult problem. IMHO, this question is too broad.
Some problems even formulating the question:


*

*What does passing a bill even mean?


*

*Bills change! If a bill is amended and then passes, does it count as the original bill passing? What if the bill is so heavily amended that is unrecognizable? 

*It's hard to even define the outcome variable!


*

*If you observe outcome $\omega$ how do you define random variable $Y(\omega)$ telling you whether the bill passed? 



*It is entirely wrong to think that Congress schedules a vote for time $t$ and then we observe the outcome like a coin flip.


*

*Let $X_t$ be the time that a vote on a bill is completed. This isn't a fixed time. It's a random variable itself, a stopping time! If the House Leadership wants a bill to pass and the bill is in danger of not passing, they will pull the bill or reschedule the vote.


*

*The problem is even worse than this because you don't know (from simply looking at votes) whether a vote is the final vote on a bill. There's always the possibility of a motion to reconsider.




Further problems...
Members of Congress are immersed in a massively complicated, game theory type problem with asymmetric information. Think of something like poker but with more cards, more players, and massive collusion between parties.
A few ideas almost certainly emerge.


*

*A theme in game theory is that controlling the order of decisions can exert massive influence on the outcome. House Leadership can force certain outcomes by eliminating alternatives preferred by some members. 

*Sycorax does a good job of discussing all kinds of voting blocks. 
If I were modeling this...
For me, my starting point would NOT be some absurd model where every Congressional vote is an independent coin flip. My starting point would be something motivated more by the median voter theorem.


*

*Can I reduce the issue to a single dimension?

*Who is the median voter in Congress along this dimension?


For example, several key aspects of the Affordable Care Act were structured so as to gain the support of Senator Ben Nelson of Nebraska as he was marginal voter that the Democrats needed to get to a filibuster proof majority of 60 votes.
If you were forecasting whether the ACA would pass, it eventually got to the point of forecasting what Ben Nelson would do. On the Supreme Court, forecasting the outcome often has been a question of what will Justice Kennedy do?
Identify swing voters, and make sure you model the extent to which they can swing together?
Disclaimer: I'm not a political science guy. Undoubtedly there are quantitative, statistical political scientists who study these types of questions. 
