Tina randomly selects two distinct numbers from the set \1, 2, 3, 4, 5\, and Sergio randomly selects a number from the set \1, 2, …, 10\. The probability that Sergio's number is larger than the sum of the two numbers chosen by Tina is

2002 AMC 12A Problem 16 — Solution

2002 AMC 12A Problem 16

Below is the professionally curated solution for Problem 16 of the 2002 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2002 AMC 12A solutions, or check the answer key.

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Concepts:basic probability casework

Difficulty rating: 1630

16.

Tina randomly selects two distinct numbers from the set $\{1, 2, 3, 4, 5\},$ and Sergio randomly selects a number from the set $\{1, 2, \ldots, 10\}.$ The probability that Sergio's number is larger than the sum of the two numbers chosen by Tina is

$\dfrac{2}{5}$

$\dfrac{9}{20}$

$\dfrac{1}{2}$

$\dfrac{11}{20}$

$\dfrac{24}{25}$

Solution:

Tina's ten pairs have sums $3, 4, 5, 5, 6, 6, 7, 7, 8, 9.$ For a sum $s,$ Sergio's number exceeds it with probability $\dfrac{10 - s}{10}.$

The corresponding values of $10 - s$ are $7, 6, 5, 5, 4, 4, 3, 3, 2, 1,$ totaling $40.$ The overall probability is $\dfrac{40}{10\cdot 10} = \dfrac{2}{5}.$

Thus, the correct answer is A.

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