Statistically speaking, how common is it for a bill to pass through the House of Representatives and then fail in the Senate (or vice versa)? And what's the historical trendline of their mutual cooperation, given that the larger states are very much underrepresented in the Senate?
Statistically speaking, how common is it for a bill to pass through the House of Representatives and then fail in the Senate (or vice versa)?
About half of the bills that passed one house were not passed in a floor vote in the other house in the 114th Congress (basically the years 2015-2016), although this percentage varies dramatically from one Congress to another. There have been some Congresses where more than 90% of bills passed by one house are enacted into law.
When a bill passed one house but not the other, the bill almost always (more than 94% of the time) dies in committee the second house, without receiving a floor vote in second house.
These statistics do not reflect that fact that most bills that are passed are passed in different forms in each house and have to be reconciled in a conference committee before receiving final passage.
The vast majority of bills and resolutions never make it out of committee in their house of origination.
Is is exceedingly common for a bill to pass in one chamber and then pass in a different form in the other chamber, requiring a conference committee to reconcile the version if they can. Indeed, this is probably the predominant process in the case of any bill of any complexity.
It is not uncommon for a bill to pass in one house but fail in the other (most recently and notably, the bill seeking to repeal Obamacare which passed in the House and failed in the Senate). And, this is particularly common when the two houses of Congress are controlled by different parties (which happens less often than one might expect - the Senate tends to be the lagging factor in political control shifts).
Some illustration, although not precisely on point, can be provided from this source available in table form back to the 93rd Congress (1973-1974). The 114th Congress (mostly 2015 and 2016 plus a few days in January 2017), is the most recent full Congress (the end results aren't knowable for the 115th because many bills that have only passed neither house or only one house so far could eventually be passed in both). In that case:
- 12,063 bills and resolutions were introduced (100% of total)
- In the case of 10,344 bills and resolution (86%), "Bills and resolutions that were introduced, referred to committee, or reported by committee but had no further action."
- There were 708 passed resolutions (6%), "Passed resolutions (for joint and concurrent resolutions, this means passed both chambers)" (these can range from appointing officers or setting the calendar for a single house, to declaring a President elected based on the electoral vote, to recognizing national hamburger day; they don't always go to the President for signature). Most passed resolutions are resolutions of a single house rather than joint/concurrent resolutions and most of those deal with "housekeeping" matters.
- 661 bills (5%) got a vote, "Bills and joint/concurrent resolutions that had a significant vote in one chamber" (this includes bills and joint/concurrent resolutions ultimately passed in both chambers) except in the case of failed legislation the vote was for passage.
- 22 bills (0%) were failed legislation, "Bills and resolutions that failed a vote on passage or failed a significant vote such as cloture, passage under suspension, or resolving differences"
- 329 bills (3%) were enacted as legislation (whether or not vetoed) "Enacted bills and joint resolutions (both bills and joint resolutions can be enacted as law)"
- 9 enacted laws (0%) were vetoed by the President.
Thus, in the 114th Congress, roughly 50% of bills that pass one house are ultimately enacted as laws (this isn't exact because concurrent resolutions can get a vote and become passed resolutions but can't be enacted as a law). Of the bills that pass one house but are not enacted as laws, about 6% fail in a floor vote rejecting it in one house or the other, while the balance die in committee or otherwise never make it to a floor vote in the second house. About 3% of enacted laws were vetoed in the 114th, a percentage that varied greatly from Congress to Congress based mostly on party alignment and Presidential personality.
In practice, most bills that become enacted laws went through a conference committee stage, but these statistics don't capture that detail.
The percentage of bills which pass one house that pass both houses varies greatly. For example, in the 106th Congress, 650 bills "got a vote" and 604 were enacted into law. Failed legislation (i.e. legislation voted down on a floor vote) is always rare, with no such bills at all from 1974 to 1980, because committees do a good job of screening out bills that will lack floor support.
There has never been a huge disparity, however, in political control between the two houses. Periods of split political party control have always involved fairly thin margins for the party in control of each chamber. The system itself doesn't inherently lead to that result, but the politics of the admission of new states has largely preserved the balance of power such that the political division in the House are roughly mirrored in the Senate.