Question

assuming a dart randomly strikes the interior region of the large square, what is the probability that the dart does not land in the shaded interior square?
$\frac{11}{16}$
$\frac{1}{4}$
$\frac{5}{8}$
$\frac{7}{16}$

Explanation:

Step1: Define grid unit length

Let each grid square side = 1 unit.

Step2: Find large square side length

Large square spans from $x=-4$ to $x=4$, so side length $= 4 - (-4) = 8$ units.

Step3: Calculate large square area

Area $A_{large} = 8^2 = 64$

Step4: Find shaded square side length

Shaded square spans from $x=-3$ to $x=3$, so side length $= 3 - (-3) = 6$ units.

Step5: Calculate shaded square area

Area $A_{shaded} = 6^2 = 36$

Step6: Calculate unshaded area

$A_{unshaded} = A_{large} - A_{shaded} = 64 - 36 = 28$

Step7: Compute target probability

Probability $= \frac{A_{unshaded}}{A_{large}} = \frac{28}{64} = \frac{7}{16}$

Answer:

$\frac{7}{16}$