Geometric Probability: Definition, Formula & Examples

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  • 0:00 What Is Geometric Probability?
  • 1:31 Geometric Probability Formula
  • 2:06 Examples
  • 4:32 Lesson Summary
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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.

What Is Geometric Probability?

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.

Geometric Probability Formula

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

Examples

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.

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