Geometric Probability Formula - Definition & Examples


Probability of an event can be defined as the likelihood of an event to occur. Suppose we have a ground of 200 square feet, and we want to throw a ball such that it hits somewhere in a specified area of 20 square feet, then what will be the probability of the ball hitting the desired area?

This probability of hitting a certain area from a given space is known as geometric probability. For the above question, the geometric probability will be $\frac{20}{200}$ = $\frac{1}{10}$.


To find the geometric probability, we need to find the total area and the desired area. Total area will be the area of given space, and the desired area will be the area where we are aiming at. The geometric probability is the ratio of desired area to total area.
$Geometric\ Probability$ = $\frac{\text {Desired Area}}{\text {Total Area}}$

A big square has a one side of 24 cm, and has a small square within it of area 16 cm square. Find the probability of a dart hitting the small square.


We have total area as the area of big square. Hence, total area = $24\ \times\ 24$ = $576$.

Desired area is given to be area of small square, that is, 16.

Geometric probability = $\frac{16}{576}$ = $\frac{1}{36}$.

Hence, the probability of hitting the small square is $\frac{1}{36}$.


Lets consider few of the examples:
Example 1:  

Suppose there is a rectangular playground with sides 15 feet and 25 feet respectively. It has a semicircle area on one of its sides with radius 5 feet. If a ball hits that area, it will be counted as a point. Find the probability of getting a point.


The total area will be the area of the rectangular field. Hence, total area = $15\ \times\ 25$ = $375$

Desired area will be the area of the semicircle.

Desired area = $\frac{\pi \times\ 5^2}{2}$ = $12.5\pi$

Geometric probability = $\frac{12.5\pi}{375}$ = $0.10467$

Hence, the probability of getting a point is $0.10467$
Example 2: 

In a circle of radius 5 cm, there is a square of side 2 cm. Find the probability that a dart thrown will hit the circle but not the square.


Total area = $\pi\ \times\ 5^2$ = $25\pi$

Area of square = $2\ \times\ 2$ = $4$

Desired area = $25\pi$ - $4$

Geometric Probability = $\frac{25\pi\ -\ 4}{25\pi}$ = $0.94904$

Hence, the probability is $0.95904$.