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SAT Mathematics Level 2: Help and Review22 chapters | 225 lessons

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Lesson Transcript

Instructor:
*Miriam Snare*

Miriam has taught middle- and high-school math for over 10 years and has a master's degree in Curriculum and Instruction.

This lesson will help you understand the concept of geometric probability. We will work through a few examples, then you can test your knowledge with a quiz.

**Probability** is a number value that shows how likely it is that some particular event will happen. With **geometric probability**, you are looking for the likelihood that you will hit a particular area of a figure. So, geometric probability is a bit like a game of darts.

Probability is always expressed as a ratio between 0 and 1 that gives a value to how likely an event is to happen. A probability of 0 means there is no chance of that event happening. For example, the probability of being bitten by a shark while walking through the desert is 0. A probability of 1 means the particular event will always happen. For example, if you jump into a lake, the probability that you will get wet is 1. A probability of 0.5 means there is a 50/50 chance of the event happening, like getting tails when you flip a coin.

All possible outcomes for a situation add up to a probability of 1. This is because we are going to assume that nothing else could happen, except for the events we are considering. So, when you flip a coin, we consider only that it could come up heads or tails. We are going to ignore the fact that the coin could land on the edge.

In this lesson, we'll be looking at playing darts as an example of calculating geometric probability. We're going to assume that the dart will land in one of the areas on the dartboard. We are going to ignore that someone could be so bad at darts that the dart misses the board completely.

To calculate geometric probability, you will need to find the areas of the shapes involved in the problem. You'll need to know the **total area**, which means the biggest area in the diagram, like the entire dartboard. You will also need to know the **desired area**, which is the part you are trying to hit, like the bull's eye.

Once you have calculated both of these areas, the formula is simply:

*P* = *desired* / *total*

In this formula, *P* stands for geometric probability*Desired* stands for the area that you want to hit*Total* stands for the area of the whole figure

Let's look at this diagram and figure out several geometric probabilities:

We are going to try to find the following:

1) The geometric probability of hitting the red circle.

2) The geometric probability of hitting the red or the green area.

3) The probability of hitting the white area.

4) The probability of hitting the yellow area.

To answer the first question, we need to calculate the desired area and the total area. In this case, the red circle is the desired area. It is a circle with a radius of 2. So, we find the desired area to be pi times two squared, which equals 4 pi.

Then, we find the total area, which is the area of the white rectangle. The rectangle has an area of 7 x 9 = 63.

The geometric probability of hitting the red circle is (4pi) / 63. That comes out to be a probability of approximately 0.199.

For the second question, the desired area is the area covered by two circles. If we find the area of a circle with a radius of 3, then that includes both the red circle and the green ring. So, our desired area is 3 squared pi, which equals 9 pi.

The total area, again, is the area of the rectangle, which we already found to be 63.

The geometric probability of hitting the red or green area is 9pi divided by 63. That comes out to be a probability of approximately 0.449.

For the third question, we need to find the probability of finding the white area. This is a very easy calculation based what we figured out in the second question. In the second question we found the probability of hitting anywhere in the circles. That probability was 0.449. Only one of two things can happen when you throw a dart at this board: either you land on white or you land on one of the circles. Together, both of those probabilities have to add up to 1. Therefore, if you subtract the probability of hitting the circles from 1, you get the probability of landing in the white space. So, the probability of hitting the white are is 1 - 0.449 = 0.551.

Finally, the last question is a bit of a trick question. There is no yellow area. Therefore, you have no chance of ever having a dart land on yellow. So, the probability of hitting the yellow area is 0.

**Probability** is a number value that shows how likely it is that some particular event will happen. **Geometric probability** is the calculation of the likelihood that you will hit a particular area of a figure. It is calculated by dividing the desired area by the total area.

The result of a geometric probability calculation will always be a value between 0 and 1. If an event can never happen, the probability is 0. If it always happens, the probability is 1. If you are trying to hit a very small area relative to the size of the whole board, then the probability is closer to 0. If you are trying to hit a large area relative to the size of the whole board, then the probability is closer to 1.

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SAT Mathematics Level 2: Help and Review22 chapters | 225 lessons

- Probability of Simple, Compound and Complementary Events 6:55
- Probability of Independent and Dependent Events 12:06
- Either/Or Probability: Overlapping and Non-Overlapping Events 7:05
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