2007 AMC 10B Problem 19

Problem 19 of 25HarderNumber Theory Counting & Probability

The wheel shown is spun twice, and the randomly determined numbers opposite the pointer are recorded. The first number is divided by

4,

and the second number is divided by

5.

The first remainder designates a column, and the second remainder designates a row on the checkerboard shown. What is the probability that the pair of numbers designates a shaded square?

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Solution

The shaded squares are those where the two remainders are both odd or both even. The first remainder is even (from the numbers

2

and

6

) with probability

\dfrac13

and odd with probability

\dfrac23.

The second remainder is even with probability

\dfrac12

and odd with probability

\dfrac12.

The probability that they share parity is

\dfrac13\cdot\dfrac12+\dfrac23\cdot\dfrac12=\dfrac12.

Thus, the correct answer is C.

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