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Applications of Probability

Probability in Length, Area, and Volume

This lesson is will use various real-life situations to explore the concepts of geometric probability. It will be shown how geometric probability can be calculated for one two and threedimensional objects.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

Calculating the Probability of Bumping Into a Person

Magdalena went to a sea resort with her family during her summer vacation. She noticed that when swimming in the pool with an area of square meters, one person occupies approximately square meters in the pool.

Swimming pool with people
If there are already people in the pool, what is the probability of Magdalena bumping into another person when she jumps into the pool?

Discussion

Geometric Probability

The geometric probability of an event is a ratio that involves geometric measures such as length, area, or volume. In geometric probability, points on a line segment, on a plane, or as part of a three-dimensional figure represent outcomes.

Probability and Length

In one-dimensional figures, the probability that a point chosen at random from lies on — the success region — is the ratio of the length of to the length of

Segment AD and BC

Probability and Area

In two-dimensional figures, the probability that a point chosen at random from a region lies in region — the success region — is the ratio of the area of region to the area of region

Triangular Region inside a quadrilateral

Probability and Volume

In three-dimensional figures, the probability that a point chosen at random from a solid lies inside a solid — the success region — is the ratio of the volume of solid to the volume of solid

Triangular Region inside a quadrilateral

Example

Probability of a Candy Falling on a Gray Square

Magdalena decorates a banana-honey cake she baked with candies. One of the candies fell on the floor and started to bounce. The floor consists of yellow and gray squares.

A candy on the floor
If there are yellow squares of square feet each and gray squares of square feet each, what is the probability of the candy falling on the gray region? Round the answer to two decimal places.

Hint

Calculate the combined areas of the yellow and gray squares of the kitchen floor. Identify the area of the success region and the area of total region.

Solution

It is given that there are yellow squares of square feet each and gray squares of square feet each. Therefore, by calculating the product of and as well as and the total areas of yellow and gray regions can be found. The sum of these values is the total area of the kitchen floor. Note that the whole area of the kitchen floor is the sample space of all the possible outcomes where the candy can fall. Since the probability of a candy falling on gray region should be calculated, the area of success region is the area of gray region. Therefore, the area of success region can be substituted with and the number of total region can be substituted with into the Probability Formula.
The probability of the candy falling on the gray area of the floor is approximately or

Example

Calculating the Probability of Being Lost in a Forest

Magdalena is exploring a hidden and unknown country and gets lost. She knows that the makeup of the country consists of great forests of approximately square kilometers each, fields of square kilometers each, and lakes of square kilometers each.

The land of the country with forests, fields and lakes
The total area of the country is square kilometers. Assuming that she is not in a lake, what is the probability of her being lost in a forest? In a field? Round the answer to two decimals.

Hint

Start by calculating the total area of the forests, fields, and lakes. Then use the Probability Formula.

Solution

First, the total areas of the forests, fields, and lakes should be calculated by multiplying the area of each geographical feature by the number of those features.

Object Number Area Total Area
Forests
Fields
Lakes
It is given that the total area of the country is square kilometers. Since it is also given that Magdalena is not in a lake, the area of total region she could be in is the difference between the total area of the country and the total area of lakes. Magdalena must be somewhere in a square kilometer area. To calculate the probability of the event of her being lost in a forest, substitute the total area of land covered by forests, for the area of success region and for the area of total region.
The probability of Magdalena being lost in a field can be calculated similarly by substituting the total land area made up of fields, for the area of success region and the same number for the area of total region.
Therefore, the probability of Magdalena being lost in a forest is approximately or while the probability of her being lost in a field is about or

Example

Calculating the Probability of Turning Successfully

Earlier geometric probabilities were calculated for two-dimensional problems. This time it will be shown how geometric probability can be used in a one-dimensional problem. Magdalena's friend Tiffaniqua was walking along a foot long path in a park. Portions of the path are bordered by a fence.

The path in the park
In places where the path does not have a fence, Tiffaniqua can go and sit on the grass. She decides to close her eyes and randomly turn. If the probability of Tiffaniqua successfully turning and finding a place to lie on the grass to admire the sky is what is the total length of the fence?

Hint

Use the formula for the geometric probability of length. Remember what the length of the success region represents.

Solution

The geometric probability formula can be used to find the total length of the fence. Since the probability of the event of turning and successfully finding a place in the grass is given, the length of success region is the length of parts without the fence. By substituting the length of total region with and with this value can be calculated.
It can be concluded that the length of the parts without the fence is feet. The parts of the path without the fence form the sample space of the event of turning successfully. By subtracting from the length of the path, the total length of the fence can be found. Therefore, the length of the fence is feet.

Example

Calculating the Probability of Different Positions of a Bubble

Next, the case in which the geometric probability of a three-dimensional object can be calculated will be presented. Two scientists are conducting an experiment in which they place a small bubble of water into a vacuum sphere.

A bubble in a vacuum sphere
Assuming that the bubble is equally likely to be anywhere within the sphere, what is the probability that it lands closer to the outside of the sphere than its center? Give an exact answer as a fraction in its simplest form.

Hint

Think of how the outcomes in which the bubble is closer to the outside than the center of the sphere can be described. Try to relate them to the radius of the sphere.

Solution

The experiment uses a sphere, a three-dimensional object, so the geometric probability formula for volume should be used. The sample space in this case is the whole region inside the sphere. The volume of total region is the same as the volume of the sphere. Let be the radius of the sphere. Its volume is then determined by the following formula. Outcomes in which the bubble is situated closer to the outside than the center of the sphere are said to be the success outcomes.

A bubble in a vacuum sphere
They can be calculated as the difference between the total volume of the sphere and the outcomes in which the bubble is situated closer to the center of the sphere. The bubble is closer to the center if it is in the sphere whose radius is half the radius of the whole sphere. The radius of this smaller sphere is This means that the volume of that sphere is as follows.
Simplify right-hand side
Now the volume of the success outcomes can be found.
Simplify right-hand side
Finally, the found volumes can be substituted into the probability formula.
Substitute values and evaluate
It can be concluded that the probability of the bubble being closer to the outside of the sphere is

Example

Calculating the Probability of the Girls Meeting at the Library

Magdalena and Tiffaniqua decided to meet at the school library before going home after school. Since the girls take different classes, they could arrive at two random times between and Magdalena and Tiffaniqua agreed to wait exactly minutes for each other to arrive before leaving.

Girls at the library
What is the probability that Magdalena and Tiffaniqua will see each other? Give the answer as a fraction in the simplest form.

Hint

Draw a graph in which the and axes represent Tiffaniqua's and Magdalena's timelines. Think of how the success region can be identified.

Solution

First, draw a graph that represents the possible times in which Magdalena and Tiffaniqua could meet. If Tiffaniqua arrives at then Magdalena must arrive no later than If Tiffaniqua arrives at Magdalena must arrive no later than and so on.

Timeline of girls' opportunities to meet

Therefore, the minute leeway shown on the graph represents their opportunity of meeting. The geometric probability formula for area will be used to calculate the probability of both girls being in that minute span.

Area of the Total Region

The total time available for the girls to meet or a sample space is represented by the square shown in the diagram. Its sides represent the minute time span from to

The total available time
The area of the total region can be calculated by substituting for into the area of a square formula.

Area of the Success Region

Analyzing the diagram, it can be noted that the area of the minute time allowance is equal to the difference between the area of the square and the areas of the top and bottom triangles. Also, the areas of the top and bottom triangles are the same, so calculating only one of them would be enough.

Two right triangles with the 15 minutes leeway in between
As can be seen, these are right triangles whose legs represent minutes. By using the area of a triangle formula, their areas can be calculated.
Substitute values and evaluate
Now, the area of the minute leeway zone can be found.
Therefore, the area of success region is

Calculating Probability

Finally, by substituting for the area of the success region and for the area of the total region, the probability of the event of the girls meeting can be calculated.
It can be concluded that the probability that Magdalena and Tiffaniqua will meet at the library is

Example

Calculating Probability of Hitting Different Regions on a Dartboard

On a standard dartboard, the diameter of the center red circle is centimeters and the diameter of the green circle around it is centimeters greater. Each rectangle has the width of centimeters. Some other lengths are given on the diagram.

Dartboard

A dart is thrown at the dartboard.

a What is the probability of the dart hitting a red region? Round the answer to decimals.
b What is the probability of the dart hitting a red or green region? Round the answer to decimals.
c What is the probability of the dart hitting a black or white region? Round the answer to decimals.

Hint

a Calculate the radii of all of the circles on the dartboard and then use them to find the areas of the circles. What is the total area of all the red regions? Calculate the areas in terms of pi.
b Use the geometric probability formula.
c Note that if a dart did not hit neither red nor green region, then it must have hit white or black region. Use the Complement Rule.

Solution

a In order to find the probability of hitting a red region with a dart, the geometric probability formula can be used.
To use the formula, first the areas of the success region — the red region — and total regions should be found. For the purposes of the solution the circles on the board can be named.
Darts board with identified circles
First, the radii of all the circles can be found. It is given that the diameter of the small red circle at the center of the dartboard is centimeters, so its radius is centimeter. The diameter of the green circle is said to be centimeters greater than the diameter of . This implies that the diameter of is centimeters and its radius is centimeters. It is given that the length between and the next ring is centimeters, so the radius of is obtained by adding and Next, by adding the width of the rectangles to the radius of it can be concluded that the radius of is centimeters. The radius of is equal to the sum of and radius of which is centimeters. Finally, the radius of is centimeters greater than Now that all the radii are known, they can be substituted into the circle area formula. First, the area of can be found. Leave the areas in terms of pi to get a precise result at the end.
Similarly, the areas of the rest of the circles can be calculated.
Circle Radius Area
By calculating the difference between the areas of and the total area of smaller red and green rectangles can be found. Since there are rectangles and they all have the same size, the area of each rectangle can be calculated by dividing by of the rectangles are red, so the total area of the red rectangles is square centimeters. Finally, the total area of the greater green and red rectangles can be found as the difference between the areas of and The area of each bigger rectangle can be obtained by dividing this value by Just as in case with smaller rectangles, there are bigger red rectangles, so the total area of the bigger red rectangles is square centimeters. The total red region on the board consists of and Therefore, the total area of the red region is the sum of these areas. The area of the success region was found to be square centimeters. The area of the total region is the area of the largest circle With this information in mind, the probability of the dart hitting a red region can be found.
The probability of the dart hitting a red region is about or
b Since the dart can hit the red or the green regions, the success region is the total area of both colors. Earlier it was found that the total areas of the small and big rectangles are and square centimeters, respectively. Also, square centimeters is the area of the center circles.
The total area of the success region is square centimeters. The area of total region is the same as before, square centimeters. Now enough information has been calculated to find the probability of hitting a red or green region.
The probability of hitting a red or green region is approximately or
c Note that if a dart did not hit neither red nor green region, then it must have hit white or black region. In other words, hitting a white or black region is a complement of the event of hitting red or green region. Therefore, using the Complement Rule, the probability of hitting white or black region can be found.
The probability of hitting a black or white region is approximately or

Closure

Finding Probability Using the Values of Area

Magdalena went to a sea resort with her family during her summer vacation. She noticed that when swimming in the pool with an area of square meters, one person occupies approximately square meters in the pool.

Swimming pool with people
If there are already people in the pool, what is the probability of the event of Magdalena bumping into another person when she jumps into the pool? Give the answer as a fraction in the simplest form.

Hint

Use the Probability Formula. What is the area of the success region?

Solution

It is given that there are people in the pool, each occupying square meters. By multiplying these values, the area occupied by all the people in the pool can be calculated. The area of the pool occupied by other swimmers is square meters. The Probability Formula can be used to calculate the probability of Magdalena bumping into another person. The area of the success region and the area of the total region should be substituted with and respectively.
It can be concluded that the probability of Magdalena bumping into a person in the pool is
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