Hamilton Lottery

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What are your chances of winning this lottery?

Context: Hamilton is a sold-out musical on Broadway. When a show is sold out, the theater will often do a lottery. They save 20 front-row seats at $10 apiece. We went to the Hamilton lottery and there were so many people….so few seats…. I wanted to calculate our chances of winning. I told the people running the lottery that I wanted to make a video for a math problem, and they were game. They even let me stand up on their platform and take pictures of their little counter-clicker device that kept track of how many people were in the lottery.

Details: 20 seats available. You write your name on a slip of paper and you can choose either 1 seat or 2 seats if you win. Almost everyone chooses 2 seats. I have never seen anyone choose 1 seat. So keep that in mind. The final count was 554 – that means 554 people wrote their names on slips of paper.

Extension:  What if you didn’t know everyone was gonna choose 2 tickets? Suppose people choose 2 tickets p% of the time. And they choose 1 ticket the other [100-p]% of the time. Write a formula for your chances of winning in terms of p.

Any other math questions you can think of?