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Week-VIII Probability Olasılık (Ch-3) Events, Sample Spaces, and Probability (Olay, Örnek Uzay ve Olasılık) Copyright © 2013 Pearson Education, Inc. All rights reserved Tanım 3- 2 Olasılık, olayların olabilirliğinin sayılarla ifadesidir. Günümüzde bütün bilim dallarında özellikle ekonomi, siyaset, spor, meteoroloji ve temel bilimlerde uygulama alanı vardır. Günlük hayatta havaya atılan paranın ne geleceği, bir zar atıldığında üste gelen yüzün 3 numaralı yüz olması, şans oyunları gibi,… olayların olasılıkları daha çok kullanılır. Probability is starting with an animal, and figuring out what footprints it will make. Statistics is seeing a footprint, and guessing the animal. 3- 3 3- 4 Definition 3- 5 Deney ve Çıktı: Bir madeni paranın havaya atılması sonucuna iki durum vardır. Üste gelen yüz tura veya yazıdır. Burada paranın atılması olayına deney, gelebilecek sonuçlara da çıktı denir. Definition 3- 6 Bir deneyde elde edilebilecek tüm çıktıların kümesine örnek uzay denir ve S ile gösterilir. Örnek uzayın çıktılarının her birine örnek nokta denir. Figure 3.1 Tree diagram for the coin-tossing experiment 3- 7 Figure 3.2 Venn diagrams for the three experiments from Table 3.1 3- 8 3- 9 Figure 3.3 Proportion of heads in N tosses of a coin 3 - 10 Figure 3.4 Experiment: invest in a business venture and observe whether it succeeds (S) or fails (F) 3 - 11 Procedure 3 - 12 Figure 3.5 Die-toss experiment with event A, observe an even number 3 - 13 Definition Bir deneyin örnek uzayının alt kümelerinden her birine olay, S örnek uzayına kesin olay, boş kümeye imkansız olay denir. 3 - 14 Definition 3 - 15 Procedure 3 - 16 Table 3.2 3 - 17 Table 3.3 3 - 18 Procedure 3 - 19 Exercises 3 - 20 3 - 21 3 - 22 3 - 23 3 - 24 3 - 25 3 - 26 3.2 Unions and Intersections Copyright © 2013 Pearson Education, Inc. All rights reserved Definition 3 - 28 Definition 3 - 29 Figure 3.7 Venn diagrams for union and intersection 3 - 30 Figure 3.8 Venn diagrams for die toss 3 - 31 3.3 Complementary Events Copyright © 2013 Pearson Education, Inc. All rights reserved Definition 3 - 33 Figure 3.9 Venn diagrams of complementary events 3 - 34 Procedure 3 - 35 Figure 3.10 Complementary events in the toss of two coins 3 - 36 3.4 The Additive Rule and Mutually Exclusive (Ayrık) Events Copyright © 2013 Pearson Education, Inc. All rights reserved Figure 3.11 Venn diagram of union 3 - 38 Procedure 3 - 39 Figure 3.12 Venn diagram of Mutually exclusive events 3 - 40 Definition 3 - 41 Procedure 3 - 42 Figure 3.13 Venn diagram for coin-toss experiment 3 - 43 Exercises 3 - 44 3 - 45 3 - 46 3 - 47 3.5 Conditional Probability Copyright © 2013 Pearson Education, Inc. All rights reserved Figure 3.14 Reduced sample space for the die-toss experiment: given that event B has occurred 3 - 49 Formula 3 - 50 Figure 3.15 Sample space for Example 3.15 3 - 51 Table 3.5 3 - 52 Table 3.6 3 - 53 3.6 The Multiplicative Rule and Independent Events Copyright © 2013 Pearson Education, Inc. All rights reserved Procedure 3 - 55 Figure 3.16 Venn diagram for finding P(A) 3 - 56 Figure 3.17 Venn diagram for finding P(B|A) 3 - 57 Figure 3.18 Tree diagram for Example 3.17 3 - 58 Definition 3 - 59 Figure 3.19 Venn diagram for die-toss experiment 3 - 60 Figure 3.20 Mutually exclusive events are dependent events 3 - 61 Procedure 3 - 62 Exercises 3 - 63 3.7 Random Sampling Copyright © 2013 Pearson Education, Inc. All rights reserved Definition 3 - 65 Table 3.7 3 - 66 Figure 3.21 MINITAB worksheet with random sample of 50 households 3 - 67 Figure 3.22 MINITAB worksheet with random assignment of physicians 3 - 68 Exercises 3 - 69 3.8 Some Additional Counting Rules (Optional) Copyright © 2013 Pearson Education, Inc. All rights reserved Figure 3.23 Tree diagram for shipping problem 3 - 71 Procedure 3 - 72 Procedure 3 - 73 Table 3.8 3 - 74 Table 3.9 3 - 75 Procedure 3 - 76 Definition 3 - 77 Exercises 3 - 78 3.9 Bayes’s Rule (Optional) Copyright © 2013 Pearson Education, Inc. All rights reserved Figure 3.24 Tree diagram for Example 3.33 3 - 80 Procedure 3 - 81 Exercises 3 - 82 3 - 83