Probability - Basic Probability Concepts The concept of...

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Unformatted text preview: Basic Probability Concepts The concept of probability is frequently encountered in The everyday communication. everyday For example, a physician may say that a patient has a 50-50 chance of surviving a certain operation. chance Another physician may say that she is 95 percent certain that Another a patient has a particular disease. patient Most people express probabilities in terms of percentages. Most probabilities percentages But, it is more convenient to express probabilities as But, fractions. Thus, we may measure the probability of the fractions Thus, occurrence of some event by a number between 0 and 1. number The more likely the event, the closer the number is to one. An The event that can’t occur has a probability of zero, and an event that is certain to occur hasHanan Mohamed Aly a probability of one. that Dr. 1 Dr. Objective Probability Classical Probability: For example, in the rolling of the die, each of the six sides is equally likely to be observed. So, the probability equally that a 4 will be observed is equal to 1/6. that Equally Likely Outcomes are the outcomes that have the same chance of occurring. The set of all possible outcomes (The universal The set) S contains N mutually exclusive and equally likely set) outcomes. outcomes. The empty set φ The contains no element. Dr. Hanan Mohamed Aly Dr. 2 The event, E is a set of outcomes in S which has a certain characteristic. The probability of the occurrence of E The equal to m/N. equal P(E) = m/N, where m is the number of outcomes P(E) which satisfy the event E. Dr. Hanan Mohamed Aly Dr. 3 Relative Frequency Probability: If some process is repeated a large number of times, If n, and if some resulting event E occurs m times, and the relative frequency of occurrence of E, m/n will be approximately equal to the probability of E. be P(E) = m/N. Probability measures the confidence that a particular individual has in the truth of a particular proposition. Subjective Probability For example, the probability that a cure for cancer will be discovered within the next 10 years. Dr. Hanan Mohamed Aly Dr. Hanan Mohamed Aly 4 Properties of Probability Given some process (or experiment) with n mutually Given exclusive events E1, E2, …, En, then exclusive 1- P (Ei) ≥ 0, i = 1, 2, … n 2- P (E1) + P (E2) + … + P (En) = 1 Relations Between Events: 1) U nion: A ∪ B means A or B. means Example: E xample: B 6 12 Let S = {1,2,3,4,5,6,7,8,9,10}, A be choosing an odd number > 2, 4 8 then A = {3,5,7,9}, P(A) = 0.4 and then B be choosing a number divisible by 3, then B = {3,6,9}, P(B) = 0.3. then Dr. ∪ Mohamed Aly A ∪ B = {3,5,6,7,9} and PDr. HananB) = 0.5. (A {3,5,6,7,9} 3 9 A 5 7 A 10 5 2) 2) I ntersection A ∩ B means A and B. means : Example: E xample: In the above example, A = {3,5,7,9}, In and B = {3,6,9}, then and A ∩ B = {3,9} and P(A ∩ B) = 0.2. {3,9} 12 4 8 B 6 3 9 5 7 A 10 (3 A means the complement of A, where means :Complement 12 4 8 B B 6 3 9 5 7 A 10 Dr. Hanan Mohamed Aly Dr. 6 1- A and B are called disjoint if A ∩ B = φ , and then P(A ∩ B) 1and = 0 and P(A ∪ B) = P(A) + P(B). Rules of Probability For example, if A is choosing an odd number < 11, A A = {1,3,5,7,9} and 9 7 5 3 1 B is choosing an even number < 11, B = {2,4,6,8,10}. {2,4,6,8,10}. Then P(A ∩ B) = 0 and P(A ∪ B) = P(A) + P(B) = 1. Then B 10 8 6 4 2 2- If A and B are not disjoint, then 2P(A ∪ B) = P(A) + P(B) - P(A ∩ B) For example, F or if A is choosing a number divisible by 5 A = {5,10} and B is choosing an even number < 11, B = {2,4,6,8,10}. {2,4,6,8,10}. Then P(A ∩ B) = 0.1 and Then Dr. Hanan D P(A ∪ B) = P(A) + P(B) - P(Ar. ∩ B)Mohamed Aly = 0.6. 1 3 9 7 A 5 10 2 4 6 8 B 7 3- P(A) + P(A`) = 1. For example, if A is choosing a number < 5, A = {1,2,3,4}, P(A) = 0.4, but A` = {5,6,7,8,9,10}, P(A’) = 0.6 Then P(A) + P(A`) = 1. 7 6 5 10 9 8 A 4 3 2 1 4- P(A) = P(A ∩ B) + P(A ∩ B`) 4- For example, F or if A is choosing a number divisible by 5 A = {5,10} , P(A) = 0.2 and B is choosing an even number < 11, A B = {2,4,6,8,10}. {2,4,6,8,10}. 1 Then P(A ∩ B) = 0.1 and Then 3 9 7 P(A ∩ B`) = 0.1. Dr. Hanan Mohamed D Then P(A) = P(A ∩ B) + r.P(A ∩ B`)Aly Then B 5 10 2 4 6 8 8 5- P(A` ∩ B`) = 1 – P(A ∪ B). P(A For example, if A is choosing a number divisible by 5 A = {5,10} , P(A) = 0.2 and B is choosing an even number < 11, B = {2,4,6,8,10}. {2,4,6,8,10}. Then P(A ∪ B) = 0.6 and Then P(A` ∩ B`) = 0.4. P(A Then P(A` ∩ B`) = 1 – P(A ∪ B). Then P(A P(A 1 3 7 9 A 5 10 2 4 6 8 B Two- way Table of Probabilities: B A `A Total P)A∩ ( B `B Total (P)A (`P)A P)S( = 1 9 P)A`∩ ( B (P)Br. Hanan Mohamed (`P)B Dr. Aly D (` P)A∩ B (` P)A`∩ B Calculating The Probability of an Event Example: :Here is the data of a sample of adults in a certain city Male (B) Diabetic (A) Normal (`A) Sum B 40 62 15 15 8 Female (`B) 8 62 70 A Sum 15 40 55 23 102 125 P(A) = 23 / 125 P(A`) = 1- P(A) = 102 / 125 P(B) = 55 / 125 P(B`) = 70 / 125 P(A ∩ B) = 15 / 125 P(A P(A ∩ B`) = 8 / 125 125 P(A`∩ B) = 40 / 125 P(A P(A` ∩ B`) = 62 / 125 P(A 62 P(A) = P(A ∩ B) + P(A ∩ B`) = 23 / 125 P(A ∪ B) = P(A) + P(B) - P(A ∩ B) = 23 / 125 + 55 / 125 – 15 / 125 = Dr. Hanan Mohamed Aly 10 Dr. = 63/125 Example: For a sample of 80 recently born children, the following table is :obtained Sex eight in Kg Boy Girl Sum (B) (G) 4 35 10 49 3 22 6 31 7 57 16 80 P(L) = 7/80 P(N) = 57/80 P(O) = 16/80 P(B) = 49/80 P(G) = P(B`) = 1 - P(B) = 31/80 P(L ∩ B) = 4/80 P(L P(N ∩ B) = 35/80 P(O ∩ B) = 10/80 P(L ∩ G) = 3/80 P(N ∩ G) = 22/80 P(O ∩ G) = 6/80 P(L ∪ B) = 7/80 + 49/80 - 4/80 = P(L = 52/80 P(N ∪ B) = 57/80 + 49/80 - 35/80 = = 71/80 P(O ∪ B) = 16/80 + 49/80 - 10/80 = P(O = 55/80 P(L ∪ N) = 7/80 + 57/80 = 64/80 P(L P(B ∪ G) = 49/80 + 31/80 = 1 P(B 11 Kg 2.5 > (L) 3 > ­ 2.5 (N) + 3 (O) Sum Dr. Hanan Mohamed Aly Dr. ...
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This note was uploaded on 09/28/2009 for the course ENGINEERIN EMP5100 taught by Professor Guy-vincentjourdan during the Fall '09 term at University of Ottawa.

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