You cannot know for certain which way an aircraft will make its final approach, or which way it will turn prior to it; local weather, especially ground level wind, is the most important factor.
Air traffic control may redirect you based on other traffic, prevailing winds may shift, and so on.
The most famous approach to DCA is the River Visual , following the Potomac River past many Washington landmarks, but depending on the wind, you may come in from the opposite direction entirely on the Mount Vernon Visual .
That said, some airports are more predictable than others.
LAX is situated on the ocean, so almost all landings are to the west into the onshore breeze (whether your flight is directed north of the airport or south to turn is subject to conditions, of course).
Depending on the airport, you may be able to look up historical and realtime flight tracks on a site like FlightAware.com , Planefinder.net , or Radarbox24.com .
From there you can make an educated guess, but there are no guarantees.
Even LAX sees landings to the east when the Santa Ana winds are blowing hard.
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Problem is most of the time you can't.
The reason being is that depending on the duration of the flight the weather condition may shift and the landing direction and the runway may shift with it.
As an example Newark Liberty Airport you can land on runways 4R/22L in either direction and sometimes even on Runway 29, the weather conditions and especially wind direction play a role and can shift in the hours that your flight is in air.
As I pass by the airport daily I can watch planes land on runway
22 Left (southerly direction) or 4 Right (northerly direction).
So the only concern is safety of the flight operations on the part of the airport management and Air Traffic Control.
The only time that you will be able to determine landing direction at the airport is when the airport has well defined wind direction during the course of a day.
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If you are willing to wait until a few hours before the flight, you often can make a prediction of which side will be the inside of the aircraft on the approach, and for takeoff- if you can predict the runway, you can figure it out by reading the standard instrument departure procedure for the flight plan that was filed (usually on flightaware.com 2-6 hours before US flights).
However, as others said it is very hard to predict which runway, as there are many details that affect it.
Generally if the winds are less than 6-8 knots, the airport will be in a calm winds configuration, which varies by airport.
For example, at Boston Logan, calm winds is depart runway 09, arrive runway 04R, unless the aircraft is a propeller plane, in which case, they depart runway
04L.
(Source: http://bostonartcc.net/document_library/sops/kbos.pdf ).
It gets much easier though at airports with one major runway.
So, at Washington National, if the winds are from the south, they will land runway 19, usually on the River Visual, if the winds are from the north, they will land and depart runway 01, which means they will turn left towards Arlington immediately after takeoff.
If you take all of this into account and get the right wind forecast, you can often choose what side to be on for landing, however, it is much harder in my experience for takeoff, because of how many departure options there are for takeoff.
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While the other answers are correct regarding needing to predict (approximate) wind direction (in order to determine which direction planes will be traveling down a given runway and which runway(s) are likely to be in use,) another important issue (that can be difficult or impossible to predict) is to determine which runway will be used by your aircraft.
Many, if not most, major commercial airports use multiple parallel runways at the same time.
Aircraft flying into or out of these runways will usually be required to bank in opposite directions in order to keep the patterns away from each other.
For instance, the Nashville Intl airport has 3 parallel runways that are used simultaneously.
Atlanta has 5 of them.
It's mostly just up to ATC to determine which runway a given aircraft will be assigned to depart from or arrive to, though a pilot may request a particular runway.
In some airports, some of the parallel runways are longer than others, so if you're on a long-haul flight, you'll be more likely to be assigned the longer runways (assuming they're in active use.)
Most domestic flights could be assigned any of the runways at most major airports, though.
Another (imperfect) method of predicting which runway you'll be assigned is to determine which way your flight is going.
For instance, if you're in Nashville and heading East, you'll be more likely to take off from 02R/20L, since its departure pattern typically turns towards the East.
On the other hand, if you're going West, you might be more likely to be assigned 02C/20C or 02L/20R. None of these are absolutes, though.
If there's a specific airport you're concerned with, you could probably ask someone who is familiar with flying in and out of that airport what the typical commercial jet traffic pattern(s) are for the runway(s) there.
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Some airports are positioned such that they have flightpaths for both left and right banking, like Schiphol Amsterdam.
Most likely the one runway will to left and the other go right, but if only one of them is active all flights might go the other direction than the usual one for that runway.
So even knowing your runway and wind direction, you can not know which way the plane will go.
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How can a layperson airline passenger find out (including in advance from wind forecasts etc) whether an airline will bank left or right, after takeoff and before landing for a given airline, time and airports?   If given a choice, I would like to choose a seat on the inside of a turn so I can have a better view of the ground and nearby towns/cities.
How to find out which way airliners bank on takeoff and landing
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Some airports  have flightpaths for both left and right banking, so unless the airport has well-defined wind direction during the course of the day, you cannot know in advance. Traffic control may redirect you based on other traffic, prevailing winds may shift, and so on. If you wait until a few hours before the flight, you often can predict the inside of the aircraft on the approach and landing based on the probable runway and the standard instrument departure procedure for the flight plan that was filed (usually on flightaware.com 2-6 hours before US flights). it is not easy to guess which side a plane will bank for landing as landing decisions are based on wind conditions. You could have a good idea on a day when wind conditions are stable throughout the day.
It is difficult to know which way a plane will bank on landing because the decision will be made depending on wind conditions, etc. Some airports, like Schiphol Amsterdam, have runways for right and left banking. You can guess which side the plane will bank if the wind conditions are stable during the day, so obviously this can only happen shortly before the flight.