Probability - musical chairs help

[MaryStew]

I have a question below and was wondering if someone could confirm my first answer and guide me through the 2nd? I think I have the 1st question figured out, but I'm having a really tough time visualizing the 2nd question.

We are pIaying a game of bIindfolded musicaI chairs with 20 bIindfoIded people and 40 chairs. When the music stops each person picks a chair uniformIy at random and sits on it.

i. What is the probabiIity that some chair has at least two people sitting on it?

The probability the first person picks an empty chair is 40/40 = 1
The probability the second person picks an empty chair is 39/40
The probability the third person picks an empty chair is 38/40
...
The probability the twentieth person picks an empty chair is 21/40

Multiplying these together give me 0.003050146 and subtracting this from 1 gives me 0.996949854.

ii. What is the probabiIity that some chair has at least three people sitting on it?

I'm really confused as to how I would do the 2nd part and was hoping for some guidance/help?

 

[MaryStew]

Like is there an efficient way to do this?

 

[Dr.Peterson]

I don't have an answer; the problem reminds me of the question of the probability that at least 3 people in a room have the same birthday, which I have only seen done in a complicated way.

My question for you is, did you make up the problem, or was it assigned to you in a context that implies there should be a reasonable way to solve it? Not all problems have nice answers ...

 

[MaryStew]

I did not makeup the problem. It was given to me. I was just wondering if there are any efficient ways of doing the 2nd question that I'm not thinking of

 

[Steven G]

Try picking a chair, say chair #1.

Now what is the probability that at least three people pick that chair? How can this information help you?

 

[RobertGytre]

I think this chair picking review solve all the problems.