Instruction 7-3

Probability of Events | Relationships Between Events | Geometric Probability | Summary

The probability of locating a random point in a particular region as a part of a whole is called geometric probability. For example, finding a measure for possibility of striking a meteor a land is geometric probability. Since in this observation, the ratio of the area of the lands to the entire area of the earth is considered as a measure for probability. In other words, in geometric probability, we always deal with the ratio of lengths, areas, or volumes.

Assume that a dart is thrown toward a rectangular board shown below, where there are two equal circles each measures 2 in. What is the probability of landing a dart outside the circles?

Let us first calculate the areas of each region. Denote the area of the rectangle by R, the area of each circle by C, and the area bounded by the rectangle and circles by B. Then,

Now, we need to find the ratio of 58.88 to 84 as the required probability:

Practice 4. The square below is a dart board. If one throws 320 darts randomly and all the darts hit the board, how many darts you expect to land on the rectangle C?

Solution
Let S be the area of the outer square, and c be the area of the rectangle C. Then

The ratio of c to A is .

Assume that the number of the darts expected to land on C is x. Then the ratio of x to the number of darts to be thrown is equal to the ratio of c to A. That is,

Cross multiply this equation.

4x = 320

Divide each side by 4.

x = 80

Thus, with equal probability for all darts thrown at the board, we expect 80 of them to land on the rectangle C.

Video Instruction
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• Solving Some Geometric Probability Problems – Length and Areas [7:27]

• Geometric Probability [9:44]

• Geometric Probability – No Words [4:06]

for Students, Parents and Teachers

Now let's do Practice Exercise 7-3-a (top).

Summary

Probability of an Event. Any subset of a sample space is called an event. If an event has one element only it is then called a simple event.
Probability of an Event in a Uniform Sample Space. If an event E results in n(E) equally likely outcomes and the sample space S has n(S) equally likely outcomes, then the probability of occurring the event E is .
Impossible Event. If an event is an empty set, then it is called the impossible event.

Certainty Event. If an event is the same as the sample space then it is called a certainty event.

How to Find the Number of Events of a Sample Space? If a sample space has n elements, then the number of different events can be observed is equal to 2ⁿ.

Mutually Exclusive Events. Two events A and B chosen from the same sample space are called mutually exclusive or disjoint if there is no common outcome for A and B. This property is denoted by .

Probability of Intersection of Two Events. If in a sample space, the events A and B have some common elements, then these elements are denoted by the set and the probability of such an event is calculated using the formula .

Complement of an Event. If all the elements of a sample space collected in two different sets such that they have no common elements, then these events are called complementary events, and each complements the other. Thus, the union of two complementary events is the same as the sample space and their intersection is .

Geometric Probability. The probability of locating a random point in a particular region as a part of a whole is called geometric probability. In geometric probability, we always deal with the ratio of lengths, areas, or volumes.
 

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