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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.
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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.
${\footnotesize P= \dfrac{\text{Length, Area, or Volume of success region}}{\text{Length, Area, or Volume of total region}}}$
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
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
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
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.
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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.
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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 |
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.
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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? {"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":"Length of Fence<span><span class=\"mwe-math-fallback-image-inline\" style=\"background-image: url('\/mediawiki\/index.php?title=Special:MathShowImage&hash=4479c00c3f9eb7516ec69be817ff5984&mode=mathml'); background-repeat: no-repeat; background-size: 100% 100%; vertical-align: -0.171ex;height: 1.343ex; width: 2.195ex;\" ><\/span><\/span>","formTextAfter":"ft","answer":{"text":["119"]}}
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.
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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. 
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.
Now the volume of the success outcomes can be found.
Finally, the found volumes can be substituted into the probability formula.
It can be concluded that the probability of the bubble being closer to the outside of the sphere is
▼
Substitute values and evaluate
SubstituteValues
Substitute values
SplitIntoFactors
Split into factors
CrossCommonFac
Cross out common factors
CancelCommonFac
Cancel out common factors
MultFrac
Multiply fractions
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.
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What is the probability that Magdalena and Tiffaniqua will see each other? Give the answer as a fraction in the simplest form. {"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":"<span><span class=\"mwe-math-fallback-image-inline\" style=\"background-image: url('\/mediawiki\/index.php?title=Special:MathShowImage&hash=94544ed87ef062a253f392a78ea14de2&mode=mathml'); background-repeat: no-repeat; background-size: 100% 100%; vertical-align: -0.838ex;height: 2.843ex; width: 14.022ex;\" ><\/span><\/span>","formTextAfter":null,"answer":{"text":["11\/36"]}}
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.
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.
The area of the total region can be calculated by substituting for into the area of a square formula.
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.
Now, the area of the minute leeway zone can be found.
Therefore, the area of success region is
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 toArea 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.▼
Substitute values and evaluate
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 isExample
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.
A dart is thrown at the dartboard.
a What is the probability of the dart hitting a red region? Round the answer to decimals.
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b What is the probability of the dart hitting a red or green region? Round the answer to decimals.
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c What is the probability of the dart hitting a black or white region? Round the answer to decimals.
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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.
| Circle | Radius | Area | |
|---|---|---|---|
SubstituteValues
Substitute values
CancelCommonFac
Cancel out common factors
SimpQuot
Simplify quotient
UseCalc
Use a calculator
RoundDec
Round to decimal place(s)
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.
SubstituteValues
Substitute values
CancelCommonFac
Cancel out common factors
SimpQuot
Simplify quotient
UseCalc
Use a calculator
RoundDec
Round to decimal place(s)
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.
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.
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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. {"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":"<span><span class=\"mwe-math-fallback-image-inline\" style=\"background-image: url('\/mediawiki\/index.php?title=Special:MathShowImage&hash=f68de2e82bb6084a077788bce2e6d4b2&mode=mathml'); background-repeat: no-repeat; background-size: 100% 100%; vertical-align: -0.338ex;height: 2.176ex; width: 4.206ex;\" ><\/span><\/span>","formTextAfter":null,"answer":{"text":["2\/5"]}}
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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