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Sampling and Probability Unit II 1 L2U1 - October 1, 2024  Definitions o Procedure or Experiment The process of obtaining an outcome that can not be predicted with certainty e.g., drawing blood for determining its typ...

Sampling and Probability Unit II 1 L2U1 - October 1, 2024  Definitions o Procedure or Experiment The process of obtaining an outcome that can not be predicted with certainty e.g., drawing blood for determining its type, tossing a coin, birth, answer to a multiple-choice question o Event Any collection of results or outcomes of a procedure e.g., with one birth, a male or a female; with blood drawn for typing, A/B/AB/O. o Probability The chance that a given event will occur. The term applies exclusively to future events, never to a past event. Sample Space Consists of all possible simple events. That is, the sample space consists of all outcomes that cannot be broken down any further e.g., with drawing blood, the sample space is A, B, AB, O Copyright © 2004 Pearson Education, Inc. 2 L2U1 - October 1, 2024  Probability Definition: Probability is the possibility that an event occur If we repeat many times an experiment, when obtained expected result, it is divided between number of experiments to know the probability If a result is sure that occur the probability will be 1 (100%) If a event is sure that does not occur the probability will be zero 3 L2U1 - October 1, 2024 Notation for Probabilities Copyright © 2004 Pearson Education, Inc. 4-1 L2U1 - October 1, 2024  Notation for Probabilities P - denotes a probability. Copyright © 2004 Pearson Education, Inc. 4-2 L2U1 - October 1, 2024  Notation for Probabilities P - denotes a probability. A, B, and C - denote specific events. Copyright © 2004 Pearson Education, Inc. 4-3 L2U1 - October 1, 2024  Notation for Probabilities P - denotes a probability. A, B, and C - denote specific events. P (A) - denotes the probability of event A occurring. Copyright © 2004 Pearson Education, Inc. 4-4 L2U1 - October 1, 2024 Classical definition: The probability P of event A is the number of outcomes that are "favorable to" A divided by the total number of possible outcomes i.e. Number of favorable outcomes P(A)= ------------------------------------------------ Number of possible outcomes Probability is a measure between 0 and 1. 5 L2U1 - October 1, 2024 Probability Limits Copyright © 2004 Pearson Education, Inc. 6-1 L2U1 - October 1, 2024  Probability Limits o The probability of an impossible event is 0. Copyright © 2004 Pearson Education, Inc. 6-2 L2U1 - October 1, 2024  Probability Limits o The probability of an impossible event is 0. o The probability of an event that is certain to occur is 1. Copyright © 2004 Pearson Education, Inc. 6-3 L2U1 - October 1, 2024  Probability Limits o The probability of an impossible event is 0. o The probability of an event that is certain to occur is 1. Copyright © 2004 Pearson Education, Inc. 6-4 L2U1 - October 1, 2024  Probability Limits o The probability of an impossible event is 0. o The probability of an event that is certain to occur is 1. o 0 ≤ P(A) ≤ 1 for any event A. o The probability of a present state or future event that is certain is 1. ○ Ex. P(Death)=1. o The probability that a event that is /or will not happen 0. ○ Ex. P(Human are Immortal) Copyright © 2004 Pearson Education, Inc. 6-5 L2U1 - October 1, 2024  Diagram of scale of Probability Probability that the child delivered Impossible event is a Boy Certainty 0% 50% 100% 7 L2U1 - October 1, 2024 Possible Values for Probabilities Figure 3-2 Copyright © 2004 Pearson Education, Inc. 8 L2U1 - October 1, 2024  Example: If we throw a coin in at once, the probability to obtain face is ½, because only we can obtain head or tail If we throw a dice once, the probability to obtain 4 is 1/6, because there are 6 sides in the dice If we have a box of 100 balls, 5 blue, 5 green, 10 orange, 10 yellow, 20 red, 20 white and 30 brown, the higher probability is to obtain a brown ball, 30/100 = 0.3= 30% 9 L2U1 - October 1, 2024  Example: I interviewed 200 patients in the outpatients' clinics, 123 of them were diabetics, what is the probability of randomly selecting a diabetic patient from the outpatient clinics? P(D)= 123/200=60.5% Among 420095 mobile users, 135 developed brain cancer, what is the probability of developing cancer in cell phone users? 135/420095= 0.0003=0.03% 10 L2U1 - October 1, 2024  Example Cont. Probability that the delivered child is a Boy P(Boy) = Boy/{Girl, Boy} i.e., P(Boy) = ½ i.e., The probability of delivering a Male child is found to be 50% 11 L2U1 - October 1, 2024  Empirical Definition: Cannot usually assign expected relative frequencies in real life. Example: Exact proportion of students in a university having blue eyes. If in a random sample of 100 students, 30 have blue eyes, then probability is 0.30 12 L2U1 - October 1, 2024  Two rules in probability: The additive rule for the occurrence of either event A or B. P(A or B) = P(A)+P(B)-P(A and B) Multiplicative rule: P (A and B) = P (A) x P (B) when events A and B are independent. 13 L2U1 - October 1, 2024  Example: Consider a couple having two children: The possible combinations for the sexes of two children with their probabilities: First child Second Child Boy (1/2) Girl (1/2) Boy (1/2) (boy, boy)1/4 (boy, girl) 1/4 Girl (1/2) (girl, boy) 1/4 (girl, girl) 1/4 14 L2U1 - October 1, 2024 Consider that both children are girls: The probability that the first child is a girl is 1/2. First child is a girl AND second child is also a girl. Hence, prob. (both are girls) = prob.(first child is a girl) X prob. (second child is a girl) = 1/2 x 1/2 = 1/4 The gender of children are independent events. 15 L2U1 - October 1, 2024  Additive rule: Example: Experimental Test Standard Test Total Disease No Disease (A) Positive 7 4 11 Total Negative (B) 3 86 89 Total 10 90 100 What is the probability that a patient is disease-free or has a negative test result? 16 L2U1 - October 1, 2024  Additive rule (cont..) P(A or B) = P(A)+P(B)-P(A and B) P(disease-free or test-negative) =(90/100)+(89/100)-(86/100) = 0.93 The correct number of persons who satisfy one or the other = 93% 17 L2U1 - October 1, 2024  Conditional Probability Probability of occurrence of two events together. P(event A and event B) = Probability of event A * p(event B when event A already happened) Or Probability of event B * p(event A when event B already happened) Notation: P(A and B) = P(A)P(B|A)=P(B)P(A|B). 18 L2U1 - October 1, 2024  Conditional Probability with Independent events When the two events are independent, that is the occurrence of one does not influence the occurrence of the other, then P(B|A)=P(B) and P(A| B)=P(A). Therefore the occurrence of the two events can be considered as P( A and B) = P(A). P(B) P(Postgraduate and female) Another Example: P(Death | smoker)=0.015 P(Death |non smokers)=0.005 The above mentioned probabilities are called as conditional probabilities, as the probability of death is conditional on being a smoker and a non smoker. 19 L2U1 - October 1, 2024 Probability example 20 L2U1 - October 1, 2024 Probability example 21 L2U1 - October 1, 2024  Exercise1: ⦿ 1. A researcher is interested in evaluating how well a diagnostic test works for detecting renal disease in patients with high blood pressure. She performs the diagnostic test on 137 patients 67 with known renal disease and 70 who are known to be healthy. ⦿ The diagnostic test comes back either positive (the patient has renal disease) or negative (the patient does not have renal disease). Here are the results of her experiment: 22 L2U1 - October 1, 2024  Test Result Truth Renal Disease Healthy Total Positive 44 10 54 Negative 23 60 83 Total 67 70 137 Calculate the following P(Test +)=? P(D)=? P(Test +|D)=? 23 L2U1 - October 1, 2024  Color Blindness Color blind Not color blind Total Male 14 186 200 Female 1 199 200 Total 15 385 400 P(Color blind) = ? P(Male)=? P(Color blind and male)=? P(Color blind | male) = ? P(Color blind| given woman)=? 24 L2U1 - October 1, 2024  Home work A math teacher gave her class two tests. 30% of the class passed both tests and 45% of the class passed the first test. What percent of those who passed the first test also passed the second test? 25 L2U1 - October 1, 2024 Exercise3 There are 200 birds in a zoo. 70 birds are male with brown eyes and 100 brids are female with brown eyes. 20 of the birds are male with blue eyes and 10 brids are female with blue eyes. Construct a contingency table. If a brids is selected at random, what is the probability that the bird is a female? A male with brown eyes? A female given that it has brown eyes? A male given that it has blue eyes? A creature with blue eyes given thst it is a female? 26 L2U1 - October 1, 2024