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p35_exercises-2_2 - 2—2 INTERPRETATIONS OF PROBABILITY 35...

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Unformatted text preview: 2—2 INTERPRETATIONS OF PROBABILITY 35 relative frequency of S is 1. Property 3 implies that if the events E1 and E2 have no outcomes in common, the relative frequency of outcomes in E, U E2 is the sum of the relative frequen- cies of the outcomes in E1 and E2. These axioms imply the following results. The derivations are left as exercises at the end of this section. Now, P(Q) = 0 and for any event E, P(E’) = 1 — P(E) For example, if the probability of the event E is 0.4, our interpretation of relative frequency implies that the probability of E’ is 0.6. Furthermore, if the event E1 is contained in the event E2, XERCISES FOR SECTION 2.2 ; «50. Each of the possible five outcomes of a random ex- :.- ' cut is equally likely. The sample space is {a, b, c, d, e}. U > A denote the event {a, b}, and let B denote the event is, d, e}. Determine the following: (a) P(A) (b) P(B) 1c) P(A’) (d) P(A U B) f' e) P(A n B) 42—5 1. The sample space of a random experiment is {a, b, c, , e} with probabilities 0.1, 0.1, 0.2, 0.4, and 0.2, respectively. ,-t A denote the event {a, b, c}, and let B denote the event {a, d, e}. Determine the following: (a) P(A) (b) 1’03) (0) P(A') (d) P(A U B) (e) P(A n B) ,2-52. Orders for a computer are summarized by the op- tional features that are requested as follows: proportion of orders no optional features 0.3 one optional feature 0.5 more than one optional feature 0.2 . (a) What is the probability that an order requests at least one optional feature? (b) What is the probability that an order does not request ‘ more than one optional feature? 2—53. If the last digit of a weight measurement is equally likely to be any of the digits 0 through 9, (a) What is the probability that the last digit is 0? P(El) 5 P(Ez) (b) What is the probability that the last digit is greater than or equal to 5? 2—54. A part selected for testing is equally likely to have been produced on any one of six cutting tools. (a) What is the sample space? (b) What is the probability that the part is from tool 1? (c) What is the probability that the part is from tool 3 or tool 5? (d) What is the probability that the part is not from tool 4? 2-55. An injection-molded part is equally likely to be ob- tained from any one of the eight cavities on a mold. (a) What is the sample space? (b) What is the probability a part is from cavity 1 or 2? (c) What is the probability that a part is neither from cavity 3 nor 4? 256. In an acid-base titration, a base or acid is gradually added to the other until they have completely neutralized each other. Because acids and bases are usually colorless (as are the water and salt produced in the neutralization reac- tion), pH is measured to monitor the reaction. Suppose that the equivalence point is reached after approximately 100 mL of a NaOH solution have been added (enough to react with all the acetic acid present) but that replicates are equally likely to indicate from 95 to 104 mL to the nearest mL. Assume that volumes are measured to the nearest mL and describe the sample space. (a) What is the probability that equivalence is indicated at 100 mL? (b) What is the probability that equivalence is indicated at less than 100 mL? (c) What is the probability that equivalence is indicated be- tween 98 and 102 mL (inclusive)? ...
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