According to Eric Weisstein's World of Mathematics, we have the following definitions.

The theoretical and empirical probabilties are expected to be very close in value, however they may not be the same. For example, if we roll five fair six-sided dice, adding the number rolled on each die, we "expect" to get 5 x 3.5 = 17.5 (since the average roll on any one die should be 3.5). To see what happens when we actually perform the experiment thousands of times, please perform the following java experiment.

Now suppose a dart is thrown at a 9m square target (see the following diagram below) in such a way that the dart is equally likely to hit one point of the square as any other. The entire square represents the sample space. Every dart thrown in the center 3m square (shaded green in the diagram) will be considered a successful outcome; this green region represents the event space. To calculate the probability of hitting the green square, we compare its area to that of the entire 9m square: thus we have our theoretical computation of the probability of a successful outcome as 9/81 = 1/9.

What is the probability that the dart will hit a specific point in the dartboard, such as the exact center? If there were a finite number of points in the board, say one billion, the odds of hitting the bullseye would be one out of a billion. Because there are infinitely many points on the dartboard, the probability is actually zero: this does not mean that it is impossible for the dart to land on a single point, as of course the dart must hit some point, and the probability of hitting that point is 0. This brings us to the following two definitions.

Now consider the following example. Bill and Yolanda want to meet at Sweet Things for an after dinner treat. They agree to arrive between between 8:00 PM and 8:30 PM; they also agree that the first one to arrive shall purchase two cones and wait for the other person to arrive. If, however, the second person does not arrive within 12 minutes, then the first person will start to eat that person's cone while continuing to wait for the friend. If they arrive at the same time, each will buy his own cone. What is the probability that each person eats no more than one ice-cream cone?

The sample space can be represented on the coordinate plane by a 30x30 square. Let x and y denote the number of minutes after 8:00 PM that Bill and Yolanda arrive at Sweet Things, respectively. If they arrive within 12 minutes of each other, then each will have to eat only one ice-cream cone. The event space (where the event is both arriving within 12 minutes of each other) is then equal to the sample space under the extra constraint

Now the area of the shaded region is the total area of the square, 30² = 900, minus the area of the two white triangles, or . Thus, the probability of a successful outcome is (900 - 324) / 900 = 0.64.

PROBLEM 0: There are only two people in Lexington who eat at Long John Silver's; a Mr. J. B. Spraggins and a Mrs. Eloise Taylor. They always arrive between 12:00 PM and 1:30 PM, and through years of practice, both have managed to streamline their daily routines such that they depart from the restaraunt exactly 15 minutes after their arrival.

a) What is the probability that Mr. Spraggins and Mrs. Taylor will see each other in the restaraunt?

b) Suppose that the city of Lexington puts a stop light in the entrance of Long John Silver's that allows entrance to the restaraunt every 5 minutes; i.e., at 8:00, 8:05, etc.. What is the probability that Mr. Spraggins and Mrs. Taylor will see each other in the restaraunt?