Definitely not impossible! It is true that draws are less frequent in amateur chess, and it is also true that draws are less frequent for faster games (how fast was the game in the show?), but even so, let's say that for a blitz game between evenly matched amateurs the draw frequency were 5%, or 1/20. If you played three games, the odds of playing three draws would be 1/8000, which while unlikely, is hardly "impossible". This is assuming that the probabilities are independent, which may not be entirely true but I think is close enough for a rough estimate.
I don't have an academic citation, but if you want some stats for online games you can look at the Lichess opening book: https://lichess.org/analysis . Normally it shows master-level games, but if you click on the settings you can chose to look at Lichess games instead, and filter by rating and time control. For blitz games at rating 1600, the draw frequency is 4-5%, depending on the opening move (the draw is the small gray stripe without a percentage, so you have to compute it by difference):
For the highest ratings, it's closer to 10% for blitz. And for classical chess, as you know, it's about 50% for top-level chess. For amateur chess I'd say maybe 10-20% but it depends on which level of amateur we are talking about. I won't cite the Lichess stats here because even though they have a "classical" category, it's really what would called rapid in over-the-board tournaments (30 min games in Lichess; OTB classical is usually 2 hours or more).