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Here are a few recommended readings before getting started with this lesson.
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.
P=Length, Area, or Volume of total regionLength, Area, or Volume of success region
In one-dimensional figures, the probability that a point S, chosen at random from AD, lies on BC — the success region — is the ratio of the length of BC to the length of AD.
In two-dimensional figures, the probability that a point S, chosen at random from a region R, lies in region N — the success region — is the ratio of the area of region N to the area of region R.
In three-dimensional figures, the probability that a point S, chosen at random from a solid C, lies inside a solid B — the success region — is the ratio of the volume of solid B to the volume of solid C.
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.
If there are 15 yellow squares of 2 square feet each and 18 gray squares of 3 square feet each, what is the probability of the candy falling on the gray region? Round the answer to two decimal places.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.
Substitute values
ba=b/6a/6
Calculate quotient
Round to 2 decimal place(s)
Magdalena is exploring a hidden and unknown country and gets lost. She knows that the makeup of the country consists of 8 great forests of approximately 1200 square kilometers each, 24 fields of 750 square kilometers each, and 76 lakes of 3 square kilometers each.
The total area of the country is 30000 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.Start by calculating the total area of the forests, fields, and lakes. Then use the Probability Formula.
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 | 8 | 1200 | 9600∣∣∣∣8⋅1200= |
Fields | 24 | 750 | 18000∣∣∣∣24⋅750= |
Lakes | 76 | 3 | 228∣∣∣∣76⋅3= |
Substitute values
Use a calculator
Round to 2 decimal place(s)
Substitute values
Use a calculator
Round to 2 decimal place(s)
Use the formula for the geometric probability of length. Remember what the length of the success region represents.
Substitute values
LHS⋅170=RHS⋅170
ca⋅b=ca⋅b
Calculate quotient
Rearrange equation
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.
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.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.
(ba)m=bmam
Calculate power
Multiply fractions
ba=b/4a/4
Vsphere=34πr3, Vcenter=6πr3
ca⋅b=ca⋅b
ba=b⋅2a⋅2
Subtract fractions
Substitute values
ba/dc=ba⋅cd
Split into factors
Cross out common factors
Cancel out common factors
Multiply fractions
Draw a graph in which the x- and y-axes represent Tiffaniqua's and Magdalena's timelines. Think of how the success region can be identified.
b=75, h=75
ca⋅b=ca⋅b
Multiply
Calculate quotient
Atotal=8100, Atriangle=2812.5
Multiply
Subtract term
Substitute values
ba=b/25a/25
ba=b/9a/9
On a standard dartboard, the diameter of the center red circle is 2 centimeters and the diameter of the green circle around it is 2 centimeters greater. Each rectangle has the width of 1.5 centimeters. Some other lengths are given on the diagram.
A dart is thrown at the dartboard.
Circle | Radius | πr2 | Area |
---|---|---|---|
C1 | 1 | π(1)2 | π |
C2 | 2 | π(2)2 | 4π |
C3 | 10 | π(10)2 | 100π |
C4 | 11.5 | π(11.5)2 | 132.25π |
C5 | 16 | π(16)2 | 256π |
C6 | 17.5 | π(17.5)2 | 306.25π |
Substitute values
Cancel out common factors
Simplify quotient
Use a calculator
Round to 3 decimal place(s)