~There are many interesting applications of probability that involve the concept of AREA.

~The most common would be one that deals with hitting a target with some sort of projectile (darts, arrows, etc.)

~Basically, we follow the same principles as in elementary probability, namely, use the basic definition & all rules that were stated to calculate the answer.

~If the targeted area & the area we want to hit can be calculated (not always easy), we can use the basic definition & form the ratio (area of section over the area of the target) to formulate the probability.

~A basic assumption is made that the person throwing the projectile hits the target area (i.e., dart hits the dart board).

~Also, the projectile used cannot land partially in the section of the target of interest & out of it simultaneously (i.e., a token tossed on a flat board (target) could land partially inside a section of the board & outside, simultaneously). The boundary of the section of interest is usually considered part of that section.

~Also, there must be a means to calculate the areas. This may be a very simple problem or a very complicated one. In more advanced problems, integral calculus is necessary to compute these. I will give an example illustrating a very simple case.

~Example:  Find the probability of hitting the "bulls-eye" on a circular dart board, if the entire dart board is 2 feet in diameter & the "bulls-eye" is 3/4 inches in diameter.

~Solution:

~First of all, we must use the same units for both diameters, so, convert 2 feet to inches.  (2 feet = 24 inches)

~Now, we must be able to calculate the areas of the entire dart board and the area of the "bulls-eye" region.

~Since both are circular, this is easy. (might not be so easy in other cases)

~The area of a circle is usually remembered by the formula, A=pr², where r represents the radius. Since the radius is half the diameter, we must divide each diameter by 2 first.

~We get, for the entire dart board, A = p(12)² and for the "bulls-eye",
A = p(3/8)².

~Now, simply divide the area in question by the area of the dart board, to get the probability.  P("bulls-eye") = (9/64) / 144 = 9/1024 = .008789.

~Important note:  This assumes that the dart is RANDOMLY tossed at the dart board with no region of interest in mind & absolutely NO skill involved.

~Skilled players will certainly have a much higher probability of hitting a "bulls-eye" since they, in essence, are reasonably close to the sections of interest, consequently, having the effect of limiting the size of the dart board (hence, reducing the bottom area number of the probability ratio).

~So, in short, the more skilled the dart player, the better his/her chances.
(an obvious observation...)