The answer entirely comes down to the size of the plane you're flying on, assuming that doctors on airplanes show up as frequently as they do in the general population in the U.S. (295 per 100,000 people). The larger the plane, the more likely it is that at least one of the other passengers is a physician.
I assumed that you're not traveling alone, but with 1 other person (and they are not a doctor). That leaves n - 2 other people to draw a doctor from, where n is the number of people on the plane. For simplicity, I also assumed the plane is full (which is often the case), so all I had to source was the number of seats on different airplane models, which I retrieved from SeatGuru. I then filtered on the largest domestic airlines, and threw in Spirit for good measure (although you might not make it to your destination if you fly that airline, regardless of whether there's a doctor on your flight).
Take the largest airplane model/seat configuration in my dataset, the Boeing 777-300ER (77W) flown by United Airlines. There are 366 seats on the plane, leaving 364 passengers that could be doctors. The probability the first person is a doctor is 0.295%, so the probability they are NOT a doctor is 99.71%. The same goes for the second person. And the next. And so on, or 0.9971^364 = 34.12%. This is the probability that ALL other people on the flight are NOT doctors. So the probability at least one of them IS a doctor is 1 - .3412 = 65.88%.
You would need at least 237 seats on the plane to have greater than a 50/50 chance of having a doctor on board.
American, Delta, and United crowd the top of this list, but they also make up most of the bottom too, as their size of aircraft vary widely. Meanwhile JetBlue, Southwest, and Spirit all show up lower, but they have far less variance in the sizes of the planes they fly.
