Question

what is the probability that a point chosen at random is in the blue region? options: \\(\frac{1}{16}\\), \\(\frac{1}{15}\\), \\(\frac{15}{17}\\), \\(\frac{15}{16}\\)

Explanation:

Step1: Assume side lengths

Let the side length of the large square be $4$ units (since the small square likely has side length $1$ for simplicity, and $4$ is a common multiple to make area calculations easy with the answer choices). Then the area of the large square $A_{large}=4\times4 = 16$ square units.

Step2: Calculate small square area

The side length of the small (white) square is $1$ unit (visually, it's a smaller square inside). So the area of the small square $A_{small}=1\times1 = 1$ square unit.

Step3: Find blue region area

The area of the blue region $A_{blue}=A_{large}-A_{small}=16 - 1=15$ square units.

Step4: Calculate probability

The probability $P$ that a random point is in the blue region is the ratio of the blue area to the total area: $P=\frac{A_{blue}}{A_{large}}=\frac{15}{16}$.

Answer:

$\frac{15}{16}$