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Section 7.2 Definition of Probability m is the relative frequency of an event E that occurs m times n after n repetitions. Note: The probability of an event is a number that lies between 0 and 1, inclusive. The ratio Example 1: A certain town has 5,690 people. A recent survey showed that 2,308 people wear eyeglasses regularly. What is the probability that a person chosen at random from this town does not wear eyeglasses regularly? If S={s 1 , s 2 ,…, s n } is a finite sample space with n outcomes, then the events {s 1 }, {s 2 },…, {s n } are called simple events of the experiment. Example 2: A fair die is cast. List the simple events and the probability distribution. Once probabilities are assigned to each of these simple events, we obtain a probability distribution. The probabilities, P(s 1 ), P(s 2 ),…, P(s n ) have the following properties: 1. 0 < P(s i ) < 1, i {1, 2, …, n} 2. P(s 1 ) + P(s 2 ) + … + P(s n ) = 1 3. P(s i s j ) = P(s i ) + P(s j ), i j and i, j {1, 2, …, n}. Example 3: The accompanying data were obtained from a survey of 500 Americans who were asked: How safe are American-made consumer products? Rating A (Very Safe) B (Somewhat Safe) C (Not too Safe) D (Not Safe at All) E (Don’t Know) Number of Respondents 174 199 57 43 27 Section 7.2 – Definition of Probability 1 Find the probability distribution associated with this experiment. A sample space in which the outcomes of an experiment are equally likely to occur is called a uniform sample space. Let S={s 1 , s 2 ,…, s n } be a uniform sample space. Then P(s 1 ) P(s 2 ) ... P(s n ) 1 n Finding the probability of an Event E 1. Determine the sample space S. 2. Assign probabilities to each of the simple events of S. 3. If E={s 1 , s 2 ,…, s k } (k n), then P(E) P(s1 ) ... P(s k ) . Example 4: A pair of fair dice is cast. What is the probability that a. The sum of the numbers shown in the uppermost is less than 3 b. At least one 2 is cast? Section 7.2 – Definition of Probability 2 Example 5: If a marbles is selected at random form an urn containing six red marble, four blue marbles and seven white marbles. What is the probability that the marble is red? Example 6: If one card is drawn from a well-shuffled standard 52-card deck, what is the probability that the card drawn is a. a club? b. a red card? c. a seven? d. the ace of spades? Section 7.2 – Definition of Probability 3