Intro to Probabiltiy theory notes for Elements Class.pptx

Note the order probability 62 1 a machine produces 20

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Note the order Probability
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62 1. A Machine produces 20% defective components. In a random sample of 6 components. Determine the probability that: (a) There will be 3 defective components (b) There will be no more than 2 defective components (c) All the components will be non defective. p = Pr of the event happening or success [0.2] q = Pr of the event not happening or failure [0.8] n = Number of tests [6] 8.19% 0.0819 0.2 0.8 3 2 1 4 5 6 def) Pr(3 3 3 Pr (≤2 def) = Pr (0 def) + Pr (1 def) + Pr (2 def) 2 4 5 6 0.2 0.8 2 1 5 6 0.2 0.8 1 6 8 . 0 def) 2 Pr( Pr (≤2 def) = 0.26214 + 0.39321 + 0.24576 = 0.9011 = 90.11% % 21 . 26 2621 . 0 8 . 0 def) 0 Pr( 6
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63 The Mean and Standard Deviation of the Binomial Distribution The mean or average of a Binomial Distribution λ = n x p The Standard Deviation SD = Distribution about the mean value Example The probability of obtaining a defective resistor is given by 1/10 In a random sample of 9 resistors what is the mean number of defective resistors you would expect and what is the standard deviation? Mean = n x p = 9 x 0.1 = 0.9 SD = √(9x0.1x0.9) = √0.81 = 0.9 q p n σ Probability
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64 Exercises 1. A Machine produces 20% defective components. In a random sample of 6 components. Determine the probability that: (a)There will be 3 defective components (b)There will be no more than 2 defective components (c)All the components will be non defective. 2. The probability of passing an examination is 0.7. Determine the Pr that out of 8 students (a) just 2 (b) more than two will pass the examination. 3. The incidence of occupational disease in an Industry is such that the workmen have a 25% chance of suffering from it. What is the probability that in a random sample of 7 workmen (a) no one (b) not more than 2 (c) at lease 3 will contract the disease. Probability
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65 2. The probability of passing an examination is 0.7. Determine the Pr that out of 8 students (a) just 2 (b) more than two will pass the examination. p = Pr of the event happening or success [0.7] q = Pr of the event not happening or failure [0.3] n = Number of tests [8] 1.00% 0.0100 0.7 0.3 2 1 7 8 pass) Pr(2 2 6 Pr (>2 pass) = 1 – [Pr (0 pass) + Pr (1 pass) + Pr (2 pass)] 2 6 7 8 0.7 0.3 2 1 7 8 0.7 0.3 1 8 3 . 0 1 ) p 2 Pr( ass Pr (>2 pass) = 1 – [0.00006561 + 0.00122472 + 0.01000188] Pr (>2 pass) = 1 – 0.01129 = 0.98871 = 98.87% Probability
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66 3. The incidence of occupational disease in an Industry is such that the workmen have a 25% chance of suffering from it. What is the probability that in a random sample of 7 workmen (a) no one (b) not more than 2 (c) at lease 3 will contract the disease. p = Pr of the event happening or success [0.25] q = Pr of the event not happening or failure [0.75] n = Number of tests [7] % 35 . 13 1335 . 0 75 . 0 ) d 0 Pr( 7 is Pr (not 2+ dis) = Pr (0 dis) + Pr (1 dis) + Pr (2 dis) 2 5 6 7 0.25 0.75 2 1 6 7 0.25 0.75 1 7 75 . 0 ) i 2 Pr( nf Pr (≤2 inf) = 0.1335 + 0.3115 + 0.3115 = 0.7565 = 75.65% Pr (≥3 inf) = 1 - Pr (≤2 inf) = 1 – 0.7565 = 0.2435 = 24.35% Probability
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67 Example: In a certain population 10% of the population is color blind. If a random sample of 25 people is drawn from
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