Calculating Probability Using Area

To calculate the probability of an event (A), you calculate this ratio:

${"mathml":"<math style=\"font-family:stix;font-size:36px;\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathsize=\"36px\"><mfrac><mrow><mi>N</mi><mi>u</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi><mo> </mo><mi>o</mi><mi>f</mi><mo> </mo><mi>o</mi><mi>u</mi><mi>t</mi><mi>c</mi><mi>o</mi><mi>m</mi><mi>e</mi><mi>s</mi><mo> </mo><mi>w</mi><mi>i</mi><mi>t</mi><mi>h</mi><mo> </mo><mi>A</mi></mrow><mrow><mi>A</mi><mi>l</mi><mi>l</mi><mo> </mo><mi>p</mi><mi>o</mi><mi>s</mi><mi>s</mi><mi>i</mi><mi>b</mi><mi>l</mi><mi>e</mi><mo> </mo><mi>o</mi><mi>u</mi><mi>t</mi><mi>c</mi><mi>o</mi><mi>m</mi><mi>e</mi><mi>s</mi></mrow></mfrac></mstyle></math>","truncated":false}$

Suppose you have two fair coins.

If you flip these coins, these are the possible outcomes.


HT	TH	TT	HH

So the probability of the two coins landing on heads is found with this ratio:

${"mathml":"<math style=\"font-family:stix;font-size:36px;\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathsize=\"36px\"><mfrac><mrow><mi>N</mi><mi>u</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi><mo> </mo><mi>o</mi><mi>f</mi><mo> </mo><mi>o</mi><mi>u</mi><mi>t</mi><mi>c</mi><mi>o</mi><mi>m</mi><mi>e</mi><mi>s</mi><mo> </mo><mi>w</mi><mi>i</mi><mi>t</mi><mi>h</mi><mo> </mo><mi>H</mi><mi>H</mi></mrow><mrow><mi>A</mi><mi>l</mi><mi>l</mi><mo> </mo><mi>p</mi><mi>o</mi><mi>s</mi><mi>s</mi><mi>i</mi><mi>b</mi><mi>l</mi><mi>e</mi><mo> </mo><mi>o</mi><mi>u</mi><mi>t</mi><mi>c</mi><mi>o</mi><mi>m</mi><mi>e</mi><mi>s</mi></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle></math>","truncated":false}$

What about situations where there are an infinite number of possible outcomes? This brings us to considering geometric probability.

With a multicolored target what is the probability that an arrow lands in a certain area? To consider these probabilities, let’s start with a simple example.

For example, suppose a penny randomly lands on this square.

What is the probability that the penny lands in the red area? Assume the penny only lands cleanly within one of the four areas.

Each of the four sections has the same area. Can you see that the probability is ¼? This brings up a basic geometric probability equation:

${"mathml":"<math style=\"font-family:stix;font-size:36px;\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathsize=\"36px\"><mi>P</mi><mi>r</mi><mi>o</mi><mi>b</mi><mi>a</mi><mi>b</mi><mi>i</mi><mi>l</mi><mi>i</mi><mi>t</mi><mi>y</mi><mo>=</mo><mfrac><mrow><mi>A</mi><mi>r</mi><mi>e</mi><mi>a</mi><mo> </mo><mi>o</mi><mi>f</mi><mo> </mo><mi>D</mi><mi>e</mi><mi>s</mi><mi>i</mi><mi>r</mi><mi>e</mi><mi>d</mi><mo> </mo><mi>O</mi><mi>u</mi><mi>t</mi><mi>c</mi><mi>o</mi><mi>m</mi><mi>e</mi><mi>s</mi></mrow><mrow><mi>T</mi><mi>o</mi><mi>t</mi><mi>a</mi><mi>l</mi><mo> </mo><mi>A</mi><mi>r</mi><mi>e</mi><mi>a</mi></mrow></mfrac></mstyle></math>","truncated":false}$

Now let’s look at a more complicated example.

Think of this as a dartboard. There are two circular areas, the large circle and the smaller one. The circles are concentric and they each have a radius as shown. For a randomly thrown dart that lands on the target, what is the probability that it lands in the red area?

For this calculate the areas of two circles, using the general formula for the area of a circle:

${"mathml":"<math style=\"font-family:stix;font-size:36px;\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathsize=\"36px\"><mi>A</mi><mo>=</mo><mi>π</mi><msup><mi>r</mi><mn>2</mn></msup></mstyle></math>","truncated":false}$

Let’s calculate the area of the red circle and the area of the large circle.

Now we calculate the ratio of the smaller circle to the large circle to find the probability.

${"mathml":"<math style=\"font-family:stix;font-size:36px;\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathsize=\"36px\"><mtable columnspacing=\"0px\" columnalign=\"right center left\"><mtr><mtd><mi>P</mi><mi>r</mi><mi>o</mi><mi>b</mi><mi>a</mi><mi>b</mi><mi>i</mi><mi>l</mi><mi>i</mi><mi>t</mi><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd><mfrac><mrow><mi>A</mi><mi>r</mi><mi>e</mi><mi>a</mi><mo> </mo><mi>o</mi><mi>f</mi><mo> </mo><mi>D</mi><mi>e</mi><mi>s</mi><mi>i</mi><mi>r</mi><mi>e</mi><mi>d</mi><mo> </mo><mi>O</mi><mi>u</mi><mi>t</mi><mi>c</mi><mi>o</mi><mi>m</mi><mi>e</mi><mi>s</mi></mrow><mrow><mi>T</mi><mi>o</mi><mi>t</mi><mi>a</mi><mi>l</mi><mo> </mo><mi>A</mi><mi>r</mi><mi>e</mi><mi>a</mi></mrow></mfrac></mtd></mtr><mtr><mtd/><mtd><mo>=</mo></mtd><mtd><mfrac><mrow><mi>π</mi><msup><mi>r</mi><mn>2</mn></msup></mrow><mrow><mn>4</mn><mi>π</mi><msup><mi>r</mi><mn>2</mn></msup></mrow></mfrac></mtd></mtr><mtr><mtd/><mtd><mo>=</mo></mtd><mtd><mfrac><mn>1</mn><mn>4</mn></mfrac></mtd></mtr></mtable></mstyle></math>","truncated":false}$

Are you surprised that each of these has the same probability?

Now look at this example.

What is the probability of a dart landing in the green area? As before, you need to find the area of the small green region. The area of the green region is the area of the circle with radius 3r/2 minus the area of the red circle.

${"mathml":"<math style=\"font-family:stix;font-size:36px;\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathsize=\"36px\"><mtable columnspacing=\"0px\" columnalign=\"right center left\"><mtr><mtd><mi>A</mi><mi>r</mi><mi>e</mi><mi>a</mi><mo> </mo><mi>o</mi><mi>f</mi><mo> </mo><mi>l</mi><mi>a</mi><mi>r</mi><mi>g</mi><mi>e</mi><mi>r</mi><mo> </mo><mi>c</mi><mi>i</mi><mi>r</mi><mi>c</mi><mi>l</mi><mi>e</mi></mtd><mtd><mo>=</mo></mtd><mtd><mi>π</mi><msup><mfenced><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac><mi>r</mi></mrow></mfenced><mn>2</mn></msup></mtd></mtr><mtr><mtd/><mtd><mo>=</mo></mtd><mtd><mfrac><mn>9</mn><mn>4</mn></mfrac><mi>π</mi><msup><mi>r</mi><mn>2</mn></msup></mtd></mtr></mtable></mstyle></math>","truncated":false}$

Now we calculate the area of just the green disk. It is found by subtracting the two areas above:

${"mathml":"<math style=\"font-family:stix;font-size:36px;\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathsize=\"36px\"><mtable columnspacing=\"0px\" columnalign=\"right center left\"><mtr><mtd><mi>A</mi><mi>r</mi><mi>e</mi><mi>a</mi><mo> </mo><mi>o</mi><mi>f</mi><mo> </mo><mi>g</mi><mi>r</mi><mi>e</mi><mi>e</mi><mi>n</mi><mo> </mo><mi>s</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></mtd><mtd><mo>=</mo></mtd><mtd><mfrac><mn>9</mn><mn>4</mn></mfrac><mi>π</mi><msup><mi>r</mi><mn>2</mn></msup><mo>-</mo><mi>π</mi><msup><mi>r</mi><mrow/></msup></mtd></mtr><mtr><mtd/><mtd><mo>=</mo></mtd><mtd><mfrac><mn>5</mn><mn>4</mn></mfrac><mi>π</mi><msup><mi>r</mi><mn>2</mn></msup></mtd></mtr></mtable></mstyle></math>","truncated":false}$

Finally we calculate the ratio of the green disk to the large circle to find the probability.

${"mathml":"<math style=\"font-family:stix;font-size:36px;\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathsize=\"36px\"><mo>÷</mo></mstyle></math>","truncated":false}$

Summary

Geometric probability is a branch of probability that deals with the likelihood of events that can be represented geometrically, such as the probability of a point falling within a certain region or the probability of two lines intersecting.

Geometric probability problems are often solved using the following formula:

For example, suppose we have a square with side length 10 cm and we randomly drop a point onto the square. What is the probability that the point will fall within a circle with radius 5 cm that is centered at the center of the square?

To solve this problem, we first need to calculate the area of the square and the area of the circle. The area of the square is 100 cm² and the area of the circle is 25π cm².

Next, we need to calculate the area of the desired outcome, which is the area of the circle.

Finally, we can calculate the probability of the desired outcome by dividing the area of the desired outcome by the total area:

Now we calculate the ratio of the green section to the area of the entire circle to the large circle to find the probability.