Question

juan and his friends played a game in which sticky darts were thrown onto a playing board. to start the game, juan threw a dart randomly onto the board. describe the likelihood of getting each score: 8 points: 16 points: more than 0 points:

Explanation:

Step1: Analyze 8 points

The 8 - point region is the smallest circle (the innermost circle). The area of this region is the smallest among the scoring regions (8, 4, 2 points) and the non - zero scoring area. So the probability of landing in the 8 - point region is very low. So the likelihood of getting 8 points is "unlikely" (or "very unlikely").

Step2: Analyze 16 points

Looking at the dartboard, there is no region marked with 16 points. So the probability of getting 16 points is 0. So the likelihood of getting 16 points is "impossible".

Step3: Analyze more than 0 points

The area of the regions with 2, 4, and 8 points (the total non - zero scoring area) is less than the area of the entire dartboard (including the non - scoring blue area), but there is a non - zero area for scoring. However, the non - scoring area (blue) exists. Wait, actually, the dart is thrown randomly. The regions with 2, 4, 8 points are the scoring regions (more than 0 points). The area of these scoring regions: the innermost circle (8), middle ring (4), outer ring (2). The total area of these is less than the area of the square (the whole board), but there is a chance to land in them. But actually, the non - scoring area is the blue part (the area of the square not covered by the largest circle). Wait, no, the largest circle is the 2 - point region? Wait, the dartboard has a square, and three concentric circles: innermost (8), middle (4), outer (2). The area of the 8 - point circle is \(A_8=\pi r_8^2\), 4 - point is \(A_4=\pi r_4^2 - \pi r_8^2\), 2 - point is \(A_2=\pi r_2^2-\pi r_4^2\), and the non - scoring area is \(A_{non - score}=s^2-\pi r_2^2\) (where \(s\) is the side length of the square, \(r_2\) is the radius of the largest circle). But to get more than 0 points, we land in 2, 4, or 8. The total area of 2 + 4+8 is \(\pi r_2^2\). The area of the whole board is \(s^2\). If the largest circle has a radius such that its diameter is equal to the side length of the square (so \(r_2=\frac{s}{2}\)), then \(\pi r_2^2=\frac{\pi s^2}{4}\approx0.785s^2\), and the non - scoring area is \(s^2 - 0.785s^2 = 0.215s^2\). So the area of more than 0 points (the area of the largest circle) is about 78.5% of the board. So the likelihood of getting more than 0 points is "likely" (since the area of the scoring regions (more than 0) is a large portion of the board, but not the entire board, but actually, if the largest circle fills most of the square, the chance is high. Alternatively, since there is a non - zero area for non - scoring, but the scoring area is big. So "likely" (or "very likely" if the circle is almost the square).

Answer:

• 8 points: Unlikely (or Very Unlikely)
• 16 points: Impossible
• more than 0 points: Likely (or Very Likely)

(Note: The exact terms can vary slightly depending on the relative areas, but based on the visual of concentric circles in a square, the 8 - point area is the smallest, 16 - point area is non - existent, and more than 0 points area is the area of the largest circle which is a significant portion of the square.)