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## Greetings at a Party

Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.

 Puzzle ID: #3406 Fun: (2.38) Difficulty: (2.67) Category: Probability Submitted By: cathalmccabe Corrected By: Winner4600

10 people are gathering in a room. Each person is meeting every other person. There are three types of greetings: a quick hello, which will take 40 seconds; a short chat, which will take 1 and a half minutes; or a longer greeting, which will take 2 minutes. Each greeting is completely random, and duration depends on the longer person. What is the probability that this gathering will take 1/2 an hour or less?
Assume no time loss between meetings with different people.

1/3^90
= 1/(8.73*10^42) !!!

All greetings need to be made, so firstly, there are 45 greetings:
9+8+7+6+5+4+3+2+1 = 45

If everyone has a short greeting:
45*40 = 1800 seconds = 30 minutes, which is ok, so if everyone made that short greeting, the probability would be 1/3^90 (Each greeting, each time depends on both people involved taking the short meeting).

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