2011 AMC 8 Problem 12

Below is the professionally curated solution for Problem 12 of the 2011 AMC 8, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2011 AMC 8 solutions, or check the answer key.

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Concepts:basic probability circular arrangements

Difficulty rating: 960

12.

Angie, Bridget, Carlos, and Diego are seated at random around a square table, one person to a side. What is the probability that Angie and Carlos are seated opposite each other?

$\dfrac{1}{4}$

$\dfrac{1}{3}$

$\dfrac{1}{2}$

$\dfrac{2}{3}$

$\dfrac{3}{4}$

Solution:

Consider that Angie's seat is chosen. Carlos has an equal probability of being in any of the other $3$ seats. Only one of them is opposite Angie, however. Therefore, the probability is $\dfrac{1}{3}.$

Thus, B is the correct answer.

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