"What are the odds?" That a Survey with 27 Choices and 1000 Participants Will End in a Tie

"We have a survey of 27 choices. There will be approximately 1,000 people voting, and they can only vote once and only make one choice each. What are the chances of a tie?"

Even without doing any math, it's obvious the number of combinations over 27 choices and 1,000 votes makes it very unlikely that there will be a tie. However, we can determine this probability indirectly using simulation.

I assumed that exactly 1,000 people will vote, and that each choice has an equal 1/27 chance of being chosen (this is obviously not likely true, but for simplicity this is the most basic assumption we can make).

I wrote a simulator in Python, and ran 10,000 simulations and tallied how often the survey resulted in a tie for first place: 14.34%.