Unlocking Geometric Probability: the Duet of Probability and Geometry

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

150

square meters, one person occupies approximately

2

square meters in the pool.

External credits: @lifeforstock

If there are already

30

people in the pool, what is the probability of Magdalena bumping into another person when she jumps into the pool?

Discussion

Geometric Probability

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.

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.

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

15

yellow squares of

2

square feet each and

18

gray squares of

3

square feet each. Therefore, by calculating the product of

15

and

2,

as well as

18

and

3,

the total areas of yellow and gray regions can be found.

Yellow : Gray : 15 \cdot 2 = 30 ft^{2} 18 \cdot 3 = 54 ft^{2}

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.

Total : 30 + 54 = 84 ft^{2}

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

54

and the number of total region can be substituted with

84

into the Probability Formula.

P (🍬 on a gray region) = \frac{Area of success region}{Area of total region}

SubstituteValues

Substitute values

P (🍬 on a gray region) = \frac{5 4}{8 4}

ReduceFrac

$\frac{a}{b} = \frac{a / 6}{b / 6}$

P (🍬 on a gray region) = \frac{9}{1 4}

CalcQuot

Calculate quotient

P (🍬 on a gray region) = 0.642857 \dots

RoundDec

Round to $2$ decimal place(s)

P (🍬 on a gray region) \approx 0.64

The probability of the candy falling on the gray area of the floor is approximately

0.64

64 % .

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 $8$ great forests of approximately $1200$ square kilometers each, $24$ fields of $750$ square kilometers each, and $76$ lakes of $3$ square kilometers each.

The land of the country with forests, fields and lakes

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.

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	$8$	$1200$	$9600 ∣ ∣ ∣ ∣ 8 \cdot 1200 =$
Fields	$24$	$750$	$18000 ∣ ∣ ∣ ∣ 24 \cdot 750 =$
Lakes	$76$	$3$	$228 ∣ ∣ ∣ ∣ 76 \cdot 3 =$

It is given that the total area of the country is

30000

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.

30000 - 228 = 29772 km^{2}

Magdalena must be somewhere in a

29772

square kilometer area. To calculate the probability of the event of her being lost in a forest, substitute

9600,

the total area of land covered by forests, for the area of success region and

29772

for the area of total region.

P (forest) = \frac{Area of success region}{Area of total region}

SubstituteValues

Substitute values

P (forest) = \frac{9 6 0 0}{2 9 7 7 2}

UseCalc

Use a calculator

P (forest) = 0.322450 \dots

RoundDec

Round to $2$ decimal place(s)

P (forest) \approx 0.32

The probability of Magdalena being lost in a field can be calculated similarly by substituting

18000,

the total land area made up of fields, for the area of success region and the same number for the area of total region.

P (field) = \frac{Area of success region}{Area of total region}

SubstituteValues

Substitute values

P (field) = \frac{1 8 0 0 0}{2 9 7 7 2}

UseCalc

Use a calculator

P (field) = 0.604594 \dots

RoundDec

Round to $2$ decimal place(s)

P (field) \approx 0.60

Therefore, the probability of Magdalena being lost in a forest is approximately

0.32,

32 %,

while the probability of her being lost in a field is about

0.60,

60 % .

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

14 : 30

P . M .

and

16 : 00

P . M .

Magdalena and Tiffaniqua agreed to wait exactly

15

minutes for each other to arrive before leaving.

External credits: @pikisuperstar, @katemangostar

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 $x -$ and $y -$ 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

14 : 30

P . M .,

then Magdalena must arrive no later than

14 : 45

P . M . .

If Tiffaniqua arrives at

14 : 31

P . M .,

Magdalena must arrive no later than

14 : 46

P . M .,

and so on.

Therefore, the

15 -

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

15 -

minute span.

P = \frac{Area of success region}{Area of total region}

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

90 -

minute time span from

14 : 30

P . M .

16 : 00

P . M . .

The area of the total region can be calculated by substituting

90

for

s

into the area of a square formula.

A_{total} = s^{2}

Substitute

$s = 90$

A_{total} = 90^{2}

CalcPow

Calculate power

A_{total} = 8100

Area of the Success Region

Analyzing the diagram, it can be noted that the area of the

15 -

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.

As can be seen, these are right triangles whose legs represent

75

minutes. By using the area of a triangle formula, their areas can be calculated.

A_{triangle} = \frac{1}{2} b h

Substitute values and evaluate

SubstituteII

$b = 75$ , $h = 75$

A_{triangle} = \frac{1}{2} (75) (75)

MoveRightFacToNum

$\frac{a}{c} \cdot b = \frac{a \cdot b}{c}$

A_{triangle} = \frac{( 7 5 ) ( 7 5 )}{2}

Multiply

A_{triangle} = \frac{5 6 2 5}{2}

CalcQuot

Calculate quotient

A_{triangle} = 2812.5

Now, the area of the

15 -

minute leeway zone can be found.

A_{success} = A_{total} - 2 A_{triangle}

SubstituteII

$A_{total} = 8100$ , $A_{triangle} = 2812.5$

A_{success} = 8100 - 2 (2812.5)

Multiply

A_{success} = 8100 - 5625

SubTerm

Subtract term

A_{success} = 2475

Therefore, the area of success region is

2475 .

Calculating Probability

Finally, by substituting

2475

for the area of the success region and

8100

for the area of the total region, the probability of the event of the girls meeting can be calculated.

P = \frac{Area of success region}{Area of total region}

SubstituteValues

Substitute values

P = \frac{2 4 7 5}{8 1 0 0}

ReduceFrac

$\frac{a}{b} = \frac{a / 2 5}{b / 2 5}$

P = \frac{9 9}{3 2 4}

ReduceFrac

$\frac{a}{b} = \frac{a / 9}{b / 9}$

P = \frac{1 1}{3 6}

It can be concluded that the probability that Magdalena and Tiffaniqua will meet at the library is

\frac{1 1}{3 6} .

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 $150$ square meters, one person occupies approximately $2$ square meters in the pool.

External credits: @lifeforstock

If there are already

30

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

30

people in the pool, each occupying

2

square meters. By multiplying these values, the area occupied by all the people in the pool can be calculated.

30 \cdot 2 = 60

The area of the pool occupied by other swimmers is

60

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

60

and

150,

respectively.

P = \frac{Area of success region}{Area of total region}

SubstituteValues

Substitute values

P = \frac{6 0}{1 5 0}

ReduceFrac

$\frac{a}{b} = \frac{a / 3 0}{b / 3 0}$

P = \frac{2}{5}

It can be concluded that the probability of Magdalena bumping into a person in the pool is

\frac{2}{5} .

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Hint

Solution

Hint

Solution

Hint

Solution

Area of the Total Region

Area of the Success Region

Calculating Probability

Hint

Solution