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 10A Problem 24 — Solution

2002 AMC 10A Problem 24

Below is the professionally curated solution for Problem 24 of the 2002 AMC 10A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2002 AMC 10A solutions, or check the answer key.

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Concepts:basic probability casework

Difficulty rating: 1900

24.

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

Averaging the winning probability over the ten pairs, the total is $\dfrac{7+6+5+5+4+4+3+3+2+1}{100}=\dfrac{40}{100}=\dfrac{2}{5}.$

Thus, the correct answer is A.

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