1995 AMC 8 Problem 20

Below is the professionally curated solution for Problem 20 of the 1995 AMC 8, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1995 AMC 8 solutions, or check the answer key.

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Concepts:dice (probability)symmetry

Difficulty rating: 1120

20.

Diana and Apollo each roll a standard die obtaining a number at random from $1$ to $6.$ What is the probability that Diana's number is larger than Apollo's number?

$\dfrac13$

$\dfrac{5}{12}$

$\dfrac49$

$\dfrac{17}{36}$

$\dfrac12$

Solution:

There are $6 \times 6 = 36$ equally likely outcomes, of which $6$ are ties, leaving $30$ outcomes with different numbers.

By symmetry, Diana is larger in exactly half of those, or $15,$ so the probability is $\dfrac{15}{36} = \dfrac{5}{12}.$

Thus, the correct answer is B .

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