Chapter 15 More About Probability

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2.1 Addition Law of Probability 2.2 Multiplication Law of Probability

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2.1 Addition Law of Probability A. Mutually Exclusive Events Definition 2.1: Two events are mutually exclusive events if they cannot occur at the same time. For two mutually exclusive events, the probability that either event happens is given by the addition law of probability.

Addition Law of Probability

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If two events E and F are mutually exclusive, then the probability that either event E or event F happens is the sum of the probabilities of events E and F, that is P(E or F) = P(E) + P(F).

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2.1 Addition Law of Probability Example 2.1 A bag contains 2 white balls, 3 red balls and 4 black balls. A ball is selected at random from the bag, find the probability that it is either white or black. Solution: Denote white balls by W1, W2, red balls by R1, R2, R3 and black balls by B1, B2, B3, B4. Possible outcomes: {W1, W2, R1, R2, R3, B1, B2, B3, B4} Content

Event of getting a white or a black ball: {W1, W2, B1, B2, B3, B4 } P(either a white or a black ball) =

6 2 = 9 3

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2.1 Addition Law of Probability Example 2.1 A bag contains 2 white balls, 3 red balls and 4 black balls. A ball is selected at random from the bag, find the probability that it is either white or black. Solution: Alternative method: Total number of balls in the bag = 2 + 3 + 4 =9 P(either a white or a black ball) = P(white) + P(black) Content

2 4 = + 9 9 2 = 3

The event of getting a white ball and the event of getting a black ball are mutually exclusive.

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2.1 Addition Law of Probability B. Complementary Events Definition 2.2: For any given event E, the event that E does not occur is called the complementary event of E. The complementary event of E is denoted by E’. Since an event either occurs or does not occur, so we have P(E) + P(E’) = 1 or P(E’) = 1 – P(E). Remarks: Content

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Event E and event E’ are complementary to each other and are mutually exclusive.

2. E is the complementary event of E’.

Two mutually exclusive events may or may not be complementary events.

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2.2 Multiplication Law of Probability A. Independent Events Definition 2.3: Two events are said to be independent if the occurrence of one event does not affect the occurrence of the other event. Example: A coin is tossed and a die is rolled. E: event of getting a tail, E = {T } F: event of getting a ‘6’, F = {6} Content

Events E and F are independent.

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2.2 Multiplication Law of Probability B. Tree Diagram When we consider two or more events, we may use a tree diagram to list all the possible outcomes. Example: If a coin is tossed twice, then we can draw a tree diagram as follows:

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In each toss, there are two possible outcomes: H stands for a head, and T stands for a tail.

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2.2 Multiplication Law of Probability Generalizing the above results, we have Multiplication Law of Probability If two events E and F are independent, then the probability that both events E and F happen is the product of the probabilities of events E and F, that is, P(E and F) = F(E) × P(F).

In general, if n events E1, E2, …, En are independent, then we have P ( E1 and E2 and ... En ) = P( E1 ) × P( E2 ) × ... × P ( En ). Content

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2.2 Multiplication Law of Probability C. Dependent Events If the probability of the second event depends on the occurrence of the first event, we call such events dependent events. Conditional Probability For any two events E and F, the probability that both E and F will happen is the product of the probability of event E and the probability of event F after event E has occurred. That is P(E and F) = P(E) × P(F after E has occurred), where P(E) > 0. Content

P(F after E has occurred) is the conditional probability of F after E has occurred. It is denoted by P(F | E), if E and F are independent, then P(F | E) = P(F).

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