SECTION 13.3 Probability 925 ‘Are You Prepared?’ Answers 1. 1; 1 2. b 65. Create a problem different from any found in the text that requires a permutation to solve. Give it to a friend to solve and critique. 66. Create a problem different from any found in the text that requires a combination to solve. Give it to a friend to solve and critique. Explaining Concepts 67. Explain the difference between a permutation and a combination. Give an example to illustrate your explanation. Problems 68–77 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for subsequent sections, a final exam, or later courses such as calculus. Retain Your Knowledge 68. Find the area of the sector of a circle of radius 4 feet and central angle θ if the arc length subtended by θ is 5 feet. 69. If ( )= − f x x2 1 and ( )= + − g x x x 2, 2 find ( )( ) g f x . 70. Give exact values for ° sin75 and ° cos15 . 71. Find the 5th term of the geometric sequence with first term = a 5 1 and common ratio = − r 2. 72. Use the binomial theorem to expand: ( ) +x y2 5 73. Solve the system: x y x y 3 4 5 5 2 17 + = − = ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ 74. Multiply, if possible: − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎡ − ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ 4 2 0 1 3 1 0 2 3 1 5 0 75. Write − +i 3 in polar form and in exponential form. 76. Find the partial fraction decomposition: + + + + x x x x 5 3 14 4 4 2 4 2 77. Write x x x 6 3 10 3 2 5 3 5 ( ) ( ) − + − as a single quotient in which only positive exponents appear. 13.3 Probability OBJECTIVES 1 Construct Probability Models (p. 925) 2 Compute Probabilities of Equally Likely Outcomes (p. 928) 3 Find Probabilities of the Union of Two Events (p. 929) 4 Use the Complement Rule to Find Probabilities (p. 930) Probability is an area of mathematics that deals with experiments that yield random results, yet admit a certain regularity. Such experiments do not always produce the same result or outcome, so the result of any one observation is not predictable. However, the results of the experiment over a long period do produce regular patterns that enable us to make predictions with remarkable accuracy. NOTE Some might say the probability that a head comes up is 50%, which isn’t wrong. However, a probability is traditionally written as a fractional value between 0 and 1, as defined on the next page. j Tossing a Fair Coin If a fair coin is tossed, the outcome is either a head or a tail. On any particular throw, we cannot predict what will happen, but if we toss the coin many times, we observe that the number of times that a head comes up is approximately equal to the number of times that a tail comes up. It seems reasonable, therefore, to assign a probability of 1 2 that a head comes up and a probability of 1 2 that a tail comes up. EXAMPLE 1 1 Construct Probability Models The discussion in Example 1 constitutes the construction of a probability model for the experiment of tossing a fair coin once. A probability model has two components: a sample space and an assignment of probabilities. A sample space S is a set whose
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