2002 AMC 10A Problem 24

Question

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}

List the ten possible sums Tina can make and how often each occurs

For a sum

s,

Sergio’s number exceeds it with probability

\dfrac{10-s}{10}

Tina’s

10

equally likely pairs give sums

3,

4,

5,

5,

6,

6,

7,

7,

8,

and

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

\small\dfrac{7+6+5+5+4+4+3+3+2+1}{100}

=\dfrac{40}{100}=\dfrac{2}{5}.

Thus, the correct answer is A.

Problem 24 in other years