Section 6.5: Conditional Probability and Independence

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Part C. Independent Events

Part A. Calculating Conditional Probability

Part B. Trees and Conditional Probability

Here is a little warm-up quiz, based on the formula for conditional probability in Part A of this tutorial.

Which of the following are correct for arbitrary events E and F?

	P(EF) = P(E|F)P(F)			P(EF) = P(E)P(F)
	P(E) = P(E|F)P(F)			P(EF) = P(F|E)P(E)

Independent Events The events E and F are independent if any one of the following three equivalent conditions hold. P(EF) = P(E)P(F). P(E|F) = P(E) (F has no effect on E) P(F|E) = P(F) (E has no effect on F) Intuitively, two events are independent if the occurrence of one has no effect on the probability of the other. If two events E and F are not independent, then they are dependent.

Example You throw two fair dice, one green and one red, and observe the numbers uppermost. E: the event that their sum is 7; P(E) = n(E)/36 = 6/36 = 1/6 F: the event that the red die shows an even number; P(F) = 1/2 P(EF) = P((1, 6), (3, 4), (5, 2)) = 3/36 = 1/12. Test for Independence P(EF) = P(E)P(F) ? 1 12 = 1 6 . 1 2 Therefore, the events are independent.

The following is similar to Example 5 on p. 449 of Finite Mathematics Applied to the Real World, or Finite Mathematics and Calculus Applied to the Real World.

You throw two fair dice, one green and one red, and observe the numbers uppermost. Which of the following pairs of events are independent?

	E: the sum is 5 F: the red die shows a 2			E: the sum is 5 F: the red die is even
	E: the sum is 5 F: the sum is 4			E: the sum is even F: the red die is even

Now try some of the exercises on pp. 453-457 of Finite Mathematics Applied to the Real World, or Finite Mathematics and Calculus Applied to the Real World.

Part A. Calculating Conditional Probability

Part B. Trees and Conditional Probability