You throw a dart at the board shown. Your dart is equally likely to hit any point inside the squ

step 1 This problem has a weird design on it. So we have a circle in the middle. We have a triangle goi

You throw a dart at the board shown. Your dart is equally...

Key Concepts

Geometric Probability

Geometric probability involves determining the likelihood of an event by comparing the measure (such as area or length) of the favorable outcomes to the measure of all possible outcomes. In problems with continuous sample spaces, such as a dartboard, the probability is calculated as a ratio of areas.

Uniform Distribution

A uniform distribution means that every point within the sample space is equally likely to be chosen. In this dartboard example, it implies that the dart has an equal chance of landing on any point within the square board.

Area Ratio

The area ratio is a method for determining probability when dealing with geometric shapes: it is the ratio of the area of the target (in this case, the yellow region) to the area of the entire sample space (the square board). This ratio directly gives the probability of the event occurring.

Transcript

00:01 This problem has a weird design on it.

00:03 So we have a circle in the middle.

00:08 We have a triangle going out to the outside, something like that.

00:14 And then it wants us to know the probability that it lands in the yellow region.

00:22 And the yellow region is the outer pieces.

00:27 And even like this part is a square where the yellow is not part of.

00:32 So the yellow, i'll just kind of label in red because the only color i have would be these larger regions on the top and bottom.

00:41 Okay, so to do probability, we need to find the total area of the dartboard.

00:44 So it's just the giant circle.

00:47 We're given that the diameter in the picture tells us that the diameter is 18 inches.

00:56 So the total area is just going to be pi r squared.

01:01 And since r would be nine in that case, it's just going to be 81 pi.

01:09 And now the yellow, it'd probably be easier since we don't know the formulas for areas of shapes like that yellow region is just going to be 81 pi minus two times the triangle area minus that square area that's around the inner circle okay so we know that's triangles are half the base times the height um and the base is going to be six inches we don't know the height yet so we'll have to do some work on that but the square on the inside is just going to be six squared so that square is 36.

01:54 Two times a triangle.

01:56 Both triangles are equivalent.

01:57 So it's just going to be base times height...

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