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Bell Work: Factor x – 6x – 16 2 Answer: (x – 8)(x + 2) Lesson 70: Probability and Designated Order The study of probability began when people began studying games of chance such as flipping coins, rolling dice, drawing cards from a deck, or drawing marbles from an urn. Problems from games of chance still provide the best models on which to base a study of elementary probability, and we will concentrate on these problems. The study of probability is based on the study of outcomes that have an equal chance of occurring. It is customary to call activities such as flipping coins, rolling dice, blindly selecting cards from a deck, and drawing marbles from an urn experiments and to call the individual results outcomes. We call the set of equally probable outcomes the sample space for the experiment. A toss of a fair coin has two equally probable outcomes. Thus, the sample space for a coin toss is heads or tails, as shown below. H T The roll of a single die has six equally probable outcomes. Thus, the figure below shows the sample space for the roll of a single die. 1 2 3 4 5 6 We define the probability of a particular even as the number of outcomes that satisfy the requirement divided by the total number of outcomes in the sample space. particular event=number outcomes that satisfy requirement total number of outcomes in sample space The probability of any event is a number between 0 and 1 inclusive. If no outcomes satisfy the requirement, the probability is 0, and if every outcome satisfies the requirement, the probability is 1. Thus we see that a probability of 2 of 7 ½ is not possible because the probability of any event must be a number between 0 and 1. Example: A fair coin is tossed three times and comes up heads every time. What is the probability that on the next toss it will come up heads? Answer: P= number of outcomes outcomes in sample space P=½ Example: Six green marbles and eight red marbles are placed in an urn. One marble is drawn and then dropped back in the urn. Then a second marble is drawn and dropped back into the urn. Both marbles were red. If another marble is drawn, what is the probability that it will be red? Answer: P = 8/14 = 4/7 Practice: A single die is rolled three times. The results are 1, 4, and 3, in that order. What is the probability that the next roll will produce a number greater than 2? Answer: 4/6 = 2/3 Practice: Two dice are rolled. What is the probability that the sum of the numbers rolled is a) 7 b) A number greater than 8 Answer: a) 6/36 = 1/6 b) 10/36 = 5/18 Designated Order: The probability of future outcomes of independent events happening in a designated order is the product of the probability of the individual outcomes. For example, if we toss a coin twice, the probability of getting a heads on the first toss and a tails on the second toss is one fourth. P(H, T) = P(H) x P(T) = ½ x ½ = ¼ Example: A fair coin is tossed four times. What is the probability that the first two times it comes up heads and the last two times it comes up tails? Answer: ½ x ½ x ½ x ½ = 1/16 Practice: The spinner show is spun twice. What is the probability that the spinner stops on 4 and then on 3? Answer: ¼ x ¼ = 1/16 HW: Lesson 70 #1-30