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.

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. 
