Are your chances of dying in a plane crash reduced if you fly direct? I recently had a disagreement with a friend about minimizing the chance of dying in a plane due to a crash. This is a rudimentary statistics question.
He stated that he prefers to fly direct to a destination, as it decreases the probability that he will die in an airplane crash. His logic was that if the probability of a commercial airline crash is 1 in 10,000, flying on two planes to get to your destination would double your chance of death.
My point was that each time one flies on an airplane, it does not increase the likelihood that he will die in a future airplane crash. That is, each airplane flight is independent. Whether someone has flown on 100 planes that year or just 1, both fliers still have a 1 in 10,000 chance of dying in a plane crash on their next flight.
Another point I made: say your destination is 4 hours away. If you take a direct flight, you will be in the air, at risk of being in a crash, for 4 hours. Now say you take 4 different connecting flights, each flight about an hour long. In this scenario you will still be in the air for roughly 4 hours. Thus, whether you take the direct flight or save some money and take connecting flights, the amount of time you spend at risk is roughly equal.
My final point was that shorter flights have a lower rate of crashes. I just pulled that one out of nowhere. I've done zero research and have zero data to back that up but...it seems logical.
Who is in the right, and why? There's a lot at stake here.
 A: Actual odds of planes crashing aside, you're falling into a logical trap here:

...each time one flies on an airplane, it does not increase the likelihood that he will die in a future airplane crash.

This is completely correct: whether you've never flown before or you've flown thousands of times, the chance of dying is still (in your example) 0.0001.
So if you're deciding between the two-hop and one-hop option, you're probably thinking about two scenarios:


*

*Future you, transferring between the two flights. Chance of dying on next flight: 0.0001.

*Future you, about to board the only flight. Chance of dying on next flight: 0.0001.


Same thing, right? Well, only if you assume you lived through the first flight in the first case. Put another way, in option 1, you're actually already dead 1/10,000th of the time. 
The general issue is that you're confusing two scenarios:


*

*your probability of being alive after $N$ flights

*your probability of being alive after $N$ flights given that you were alive after $N-1$ flights.


Your chances of surviving one flight are always $1 - 0.0001$, but overall, the chances of living to the end of $N$ flights are $(1 - 0.0001)^N$

The Opposition View: I tried to keep my answer on topic by pointing out the logical issue rather than digressing into the empirical ones. 
That said, in this case we may be letting the logic obscure the science. If your friend actually believes that skipping one flight will save him from a 1 in 10,000 chance of dying in a plane crash, the debate could be framed differently:


*

*Your statement: a two-hop flight gives you a 0.0001 chance of dying

*His statement: a two-hop flight gives a 0.0002 chance of dying


If this is the debate, it turns out that you are more correct. The actual odds of dying in a plane crash are about 1 in 2 million in the worst case. So you're both completely wrong, in that your estimates of airline fatalities are crazy high, but he's about twice as wrong as you are.
This 1 in 2 million figure is, of course, very rough and likely an overestimate. It's approximately correct to assume constant chances of dying per flight because (as many have pointed out) most accidents happen on takeoff and landing. If you really want the details, there's a lot more detail in another answer.
Condensed version: Your friend is right about probability theory, but given the statistics he's crazy to modify his behavior. 
A: Simple answer. You are correct in assuming the probability is the same for each flight, but when you make a connection you're effectively "rolling the dice" again. 
Furthermore, it's commonly known that the most dangerous points of any flight are take off and landing—thus taking a connection exposes yourself to these risks a second time. 
A: Your friend is right (on the probability theoretical side, not in in practice).
Apply your logic to throwing dice: you are saying that the chances of not throwing   snake eyes once in 100 throws (i.e. surviving 100 flights) are the same as those of not throwing snake eyes in one throw (i.e. surviving one flight). If you really think that, then I'm really interested in playing dice with you.
A: Not only do you spend more time in-flight when you have two flights to your destination, even if the layover is collinear as the crow flies (since you will interrupt cruising speed), the greatest likelihood of accidents is in take-off and landing. 
A: I'm going to answer all your questions. No, theory, all numbers.

My point was that each time one flies on an airplane, it does not
  increase the likelihood that he will die in a future airplane crash.
  That is, each airplane flight is independent. Whether someone has
  flown on 100 planes that year or just 1, both fliers still have a 1 in
  10,000 chance of dying in their next plane crash.

This might be true as a standalone fact: independence of each crash occurrence. However, it's difficult to apply to a real life. 
First, he probably meant to compare a frequent flyer vs. occasional flyer. If I fly a plane a couple of times a year to go to vacation, and his work involves weekly travel across the country, you must agree that he has a higher a chance of dying in a plane crash in the next year. We're not talking about one single flight, it's a lifestyle argument, or sample size in statistics.
Second, he's probably signed up to a frequent flyer program, which means that he always flies the same airline. Hence, the probability of a plane crash is probably more correlated in his case than in mine. So, the independence assumptions that you made is much weaker than it sounded first.
So, your friend is probably right.

Another point I made: say your destination is 4 hours away. If you
  take a direct flight, you will be in the air, at risk of being in a
  crash, for 4 hours. Now say you take 4 different connecting flights,
  each flight about an hour long. In this scenario you will still be in
  the air for roughly 4 hours. Thus, whether you take the direct flight
  or save some money and take connecting flights, the amount of time you
  spend at risk is roughly equal.

In 4 flights your cruising time is roughly the same as in 1 long flight, but you have 4 times more take-offs and descents. According to this web site, cruising is responsible for only 16% of fatalities. This graph shows the stats. You'll have more chance of dying in 4 short flights than in 1 long.


My final point was that shorter flights have a lower rate of crashes.
  I just pulled that one out of nowhere. I've done zero research and
  have zero data to back that up but...it seems logical.

This is probably not true. The shorter flights are more likely to be commuter flights, and these definitely have higher rates of fatalities according to this paper: 

Throughout the period 1977–1994, scheduled commuter flights had far
  higher crash rates than major airlines

Here, you can find some stats too. Look at the table "Which type of flying is safer" rows with Part 135 vs. 121.
If you're taking shorter flights with major airlines (which is less likely), there's still an argument on per mile basis. Per mile, the shorter flight must have higher fatality because, as I showed earlier, because you need to takeoff and land more times per mile, and these phases are the most dangerous in terms of fatalities.
UPDATE:
@AE question on what is not onboard fatalities. See this Boeing presentation with a ton of interesting data on airline crashes, where the airline accident is defined on p.3 as:

Airplane Accident: An occurrence associated with the operation of an
  airplane that takes place between the time any person boards the
  airplane with the intention of flight and such time as all such
  persons have disembarked

then the external fatalities are defined on p.4 as:

External fatalities include on-ground fatalities as well as fatalities
  on other aircraft involved.

The onboard means that death occurred to a passenger while he/she was onboard, see also CDC's reporting guideline here.
A: Your chances of a coin flip coming up heads or tails are 50/50, on EVERY flip.
However, it is highly unlikely you would ever get a run of 10 heads in a row.
A little mathematics can be a dangerous thing :-)
A: An intuitive way of looking at this, in my opinion, is the concept of micromort (one-in-a-million probability of death).
According to Wikipedia, you'll 'accumulate' roughly one micromort due to accidents for each 1000 miles traveled and roughly one micromort due to terrorism for every 12'000 miles (in the U.S.). 
This implicitly assumes that the probability of a fatal incident is proportional to the number of miles flown, independent of the number of takeoffs and landings. You'll likely get an opinion on how justified this is on https://aviation.stackexchange.com/ .
A: I believe that you and your friend have missed one important variable. That is are the chances of dying in a plane crash disproportionately concentrated in the takeoff and landing. Off the top of my head, I believe the answer is yes.
Your friend's argument is, if you fly direct for 1000 miles, versus flying two flights for 500 miles each, you've flown the same 1000 miles in each case, and therefore your chances of dying are comparable.
Your version (of my argument) would be something like "if you fly direct, you've taken off and landed once, whereas the other way, you've taken off and landed twice." If my premise (about take off and landing) is correct, the second way is almost twice as dangerous as the first. And even if my premise is wrong, that's the question you should ask.
A: 
He stated that he prefers to fly direct to a destination, as it decreases the probability that he will die in an airplane crash. 

If your friend is genuinely concerned about this incredibly low probability then they should not be flying at all, or, for that matter, driving to the airport.

His logic was that if the probability of a commercial airline crash is 1 in 10,000, flying on two planes to get to your destination would double your chance of death.

This is correct. I propose a game. Here are your choices:
Option 1 You flip a coin. Heads I win, tails, you win.
Option 2 You flip a coin. Heads I win, tails, you flip the coin again, heads, I win, tails, you win.
Me "winning" is you dying in a plane crash, you "winning" is you survive. Option 1 is you taking a single flight, option 2 is you attempting to take two flights, but you might only end up taking one if the first one crashes. 
Are the two options the same in terms of their likely outcomes, or different? Which would you pick if I gave you this game? If we were betting money on it, what would be fair odds for me to give you?

My point was that each time one flies on an airplane, it does not increase the likelihood that he will die in a future airplane crash. 

Correct. Each time you fly it decreases the likelihood that you will die in a future crash because you might die in the current flight, and so there will be no future flight to die in! 

That is, each airplane flight is independent. Whether someone has flown on 100 planes that year or just 1, both fliers still have a 1 in 10,000 chance of dying in their next plane crash.

Sure, but you had to survive those 100 flights.  
In my game, suppose you choose option two. You flip tails. You are saying "the next one is still 50-50", but you should be saying "if I'd chosen option 1 and flipped that tails, I'd be safe now, instead of once again in danger of flipping heads".  When you get off that plane having lived, you lived. Had you gotten off dead, you wouldn't be at any risk of dying on the following flight.

If you take a direct flight, you will be in the air, at risk of being in a crash, for 4 hours. Now say you take 4 different connecting flights, each flight about an hour long. In this scenario you will still be in the air for roughly 4 hours. Thus, whether you take the direct flight or save some money and take connecting flights, the amount of time you spend at risk is roughly equal.

No, that is empirically false. The vast majority of fatal commercial air disasters occur at takeoff or landing, and there is one of those per leg. Being in the air for four hours is only slightly more risky than being in the air for one, but four takeoffs are much riskier than one takeoff.
A: You have to ask yourself what this “probability of dying in plane crash” represents and how it applies to your problem (or not).
Look at it this way:


*

*If you never fly at all, your chances to die in a plane crash are going to be much lower (but not nil, cf. El Al Flight 1862 or Air France Flight 4590).

*If you fly every day for most of your life, your chances to die in a plane crash are obviously higher than if you don't.


It seems to me that there is plausible model of how plane crash deaths happen that make them completely unrelated to being in an airplane.
Consequently, I would guess that this 1 in 10,000 figure does not quite mean what you think it means. Maybe it's an average based on comparing deaths in plane crash to other causes of death or it's a reasonable estimate of the risk based on a typical “flying profile” in your country but it can't possibly be exactly the same for people with widely different plane-taking behaviour.
The difference between one flight and two flights might be tiny and your scenario also invites thinking about the difference between one flight (one take-off/one landing) and time spent in the air but if you do accept that people who never fly have a lower risk of dying in a plane crash than people who fly constantly, you can't assume than the probably of dying in a plane crash in completely independent from the number of flights you take.
