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The sample space

Example

Let event G = taking a math class. Let event H = taking a science class. Then, G AND H = taking a math class and a science class. Suppose P(G) = 0.6, P(H) = 0.5, and P(G AND H) = 0.3. Are G and H independent?

If G and H are independent, then you must show ONE of the following:

• P(G|H) = P(G)

• P(H|G) = P(H)

• P(G AND H) = P(G)P(H)

NOTE

The choice you make depends on the information you have. You could choose any of the methods here because you have the necessary information.

a. Show that P(G|H) = P(G).

Solution

P(G|H) = = = 0.6 = P(G)

b. Show P(G AND H) = P(G)P(H).

Solution 3.9 P(G)P(H) = (0.6)(0.5) = 0.3 = P(G AND H)

Since G and H are independent, knowing that a person is taking a science class does not change the chance that he or she is taking a math class. If the two events had not been independent (that is, they are dependent) then knowing that a person is taking a science class would change the chance he or she is taking math. For practice, show that P(H|G) = P(H) to show that G and H are independent events.

3.9 In a bag, there are six red marbles and four green marbles. The red marbles are marked with the numbers 1, 2, 3, 4, 5, and 6. The green marbles are marked with the numbers 1, 2, 3, and 4.

• R = a red marble

• G = a green marble

• O = an odd-numbered marble

• The sample space is S = {R1, R2, R3, R4, R5, R6, G1, G2, G3, G4}.

S has ten outcomes. What is P(G AND O)?