Question

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?

$\dfrac{1}{3}$

$\dfrac{4}{9}$

$\dfrac{1}{2}$

$\dfrac{5}{9}$

$\dfrac{2}{3}$

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.

2007 AMC 10B Problem 19

Solution:

Problem 19 in Other Years