What is the probability that a randomly thrown dart hits a blue or yellow region, given that it hits the square board? the radii of the circles are 1, 1.5, 2, 2.5, and 3 feet, respectively, and the square board has a side length of 10 feet.

a.
0.1963
b.
0.21
c.
0.2826
d.
0.3125

You are giving five radii (1, 1.5, 2, 2.5 and 3 feet) but only two colors, so I do not know which radii are for colors blue and yellow.

I will explain the procedure in detail, and so you can apply it knowing the radii of each color.

The probability of heating any circle equales the ratio of its are and the area of the square board.

The area of the square board is 10feet * 10 feet = 100 ft^2.

The area of any circle is π*(r^2).

So you can make this table

Color      radius   area                  ratio of area of the circle
                                                     to area of the square board

blue        1.0     π(1^2) = 3.14            3.14 / 100 = 0.0314

yellow     1.5     π(1.5^2) = 7.07         7.07/ 100 = 0.0707

x             2.0     π(2^2) = 12.57          12.57/100 = 0.1257

y            2.5      π(2.5^2) = 19.63       19.53 / 100 = 0.1953

z            3.0      π(3^2) = 28.27          28.27 / 100 = 0.2827

Then you have to sum the ratios for the colors blue and yellow.

In the case shown in the table that is 0.0314 + 0.0707 = 0.1021

I realize that if I sum the ratios for the second and third circles in the table, the result is 0.0707 + 0.1257 = 0.1964, which is the option A of the list.

So if the blue and yellow circles are those with radii 1.5 and 2 the answer is the option A.

With that explanation you are able to find the probability requested.