PHP2500: Introduction to Biostatistics Lecture III: Introduction to Probability 1
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Example: 40% of adults aged 40-74 in Rhode Island have pre-diabetes, a condition that raises the risk of type 2 diabetes, heart disease and stroke a. If you randomly run into an RI adult whose age is between 40-74, what is your guess about his/her pre-diabetes status? From the population of adults age 40-74 in Rhode Island, if we randomly sample 5 individuals, how many of them could have pre-diabetes? which number is most probable? how about other possible outcomes? Are they equally probable? a http://www.health.ri.gov/disease/diabetes/statistics.php 3
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Experiments and Events Probability theory makes predictions about experiments whose outcomes depend on chance. Outcome: The result of a single trial or an experiment. Event: A collection of one or more outcomes. Example: 40% of adults aged 40-74 in Rhode Island have pre-diabetes, a condition that raises the risk of type 2 diabetes, heart disease and stroke. Experiment 1: From the population of adults age 40-74 in Rhode Island, randomly sample 5 individuals Possible outcomes: none of them have pre-diabetes 1 has pre-diabetes and 4 do not have 2 have pre-diabetes and 3 do not have...... 5
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Or, if we use X to represent number of people having pre-diabetes (among the 5 chosen individuals), the possible outcomes are integers between 0 and 5. Since the value X takes is random, we call it a random variable. You may ask: what is the probability that none of them have pre-diabetes (X=0)? An event is a collection of outcomes. For example, at least two of these people have diabetes is an event that includes four outcomes: 2, or 3, or 4, or 5 of these people have pre-diabetes (X 2) Another event: less than 3 of these 5 people have diabetes ( X < 3,i.e, X 2, i.e., X=0 or X=1 or X=2,). 7
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The collection of all possible outcomes is referred to as the sample space and often represented by the symbol Ω. In this example, Ω = {0, 1, 2, 3, 4, 5} An outcome is an element of the sample space, usually represented by symbol ω An event, E, is a collection of one or more outcomes, and a subset of Ω. For example, E 1 = {1}, E 2 = {0, 1, 2} (observing 2 or less), E 3 = {4, 5} (observing 4 or more) are three different events. The events that include a particular outcome, such as E 1 in the above example, is called a simple event or an elementary event. The empty set is referred to as the null event. 9
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We have defined the set of all possible outcomes of the experiment, i.e., all possible values of X, as the sample space Ω. Any value of X must one element of Ω. We can compute the probabilities of each outcome using probability theory on Binomial distribution, which will be introduced later. For now, we will take these as given: Probability for all possible outsomes in Experiment 1. X 0 1 2 3 4 5 Prob.078.259.346.230.077.01 P (X = 3) = P (Ω) = P (0 X 5) = P ( ) = 11
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Probabilities of Experiment 1: X 0 1 2 3 4 5 Prob.078.259.346.230.077.01 Let Event 1 be either 0 or 1 person among the 5 have pre-diabetes P (E 1 ) = P (X = 0 OR X = 1) = Let Event 2 be at least 4 people among the 5 have pre-diabetes P (E 2 ) = P (X 4) = Let Event 3 be X 1 OR X 4 P (E 3 ) = In the above examples, we are combining events that are mutually exclusive. To compute probability for more complicated events, we will need some set operations. 13
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Simple set operations and notation: Intersection: A B is the set of elements that are common in A and B. If there is nothing in common, the intersection is the empty set. A = X 1 = {0, 1}, B = X 4 = {4, 5}, C= 1 X 4 A B = A C= B C= If there is no common element in two events E 1 and E 2, i.e. E 1 E 2 =, E 1 and E 2 are mutually exclusive (E 1 and E 2 cannot both occur) In the above example, A and B are mutually exclusive. A and C are not mutually exclusive (1 is common), B and C are not mutaully exclusive. P (A B) = 0 P (A C) = P (X = 1) = 0.259 P (B C) = P (X = ) = 15
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Union: A B is the set that includes all elements in A or in B (or both). Example 1: A = {1, 2, 3} P (A) =.259 +.346 +.230 =.835 B = {4, 5} P (B) = 0.077 + 0.01 = 0.087 A B = {1, 2, 3, 4, 5} P (A B) = P ({1, 2, 3, 4, 5}) = 0.922 P (A B) = P (A) + P (B) Example 2: A = {1, 2, 3} P (A) =.259 +.346 +.230 =.835 B = {3, 4, 5} P (B) =.317 A B = {1, 2, 3, 4, 5} P (A B) = P ({1, 2, 3, 4, 5}) = 0.922 P (A) + P (B) = 1.152 P (A B) What is the problem here? Notice that A B X 0 1 2 3 4 5 Prob.078.259.346.230.077.01 17
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In Example 2, A = {1, 2, 3}, B = {3, 4, 5}, the outcome X = 3 is an element of both A and B. In simple addition P (A) + P (B), this event is counted twice. We need to subtract what is double counted. P (A B) = P (A) + P (B) P (A B) Special case: For mutually exclusive events, P (A B) = P (A) + P (B), because now P (A B) = P ( ) = 0. Let s verify example 2: since A B = 3, P (A B) = 0.230 P (A) + P (B) P (A B) =.317 +.835.230 =.922 X 0 1 2 3 4 5 Prob.078.259.346.230.077.01 19
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Complement A c = Ā is the set of everything (in the sample space) that is not in A. For Ω = {0, 1, 2, 3, 4, 5}, A = {0, 1, 2, 3}, A c = {4, 5} B= at most 1 person has pre-diabetes, B c = X > 1 = {2, 3, 4, 5} C= at least 3 people have pre-diabetes, C c = D= exactly 2 people have pre-diabetes, D c = P (A c ) = 1 P (A) Exercise: compute the probability that at least one person has pre-diabetes (X 1) X 0 1 2 3 4 5 Prob.078.259.346.230.077.01 21
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For random events If intersection (A B) happens, A AND B both happen If union A B happens, A OR B OR both happen. (At least one, but both happening is possible too). Complement set A c is the event that A does not happen. A A c =? 23
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Experiment 2: 12 men and 8 women visited a clinic today. Among them, 3 men and 1 women had diabetes. Female Male Diabetes 1 3 4 No Diabetes 7 9 16 8 12 20 If one person is randomly selected from these 20, A= this person is female B= this person has diabetes What are A B and A B? P (A) = P (B) = P (A B) = P (B A) = P (A c ) = 25
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Probability rules: For any event A, 0 P (A) 1. P (Ω) = 1 = 100%. P (A) = 1 P (A c ) The probability that event A happens is the 1- the probability that A does not happen. Also, since P (Ω) = 1, P ( ) = 1 P (Ω) = 0 Independence: Two events A and B are independent if and only if P (A B) = P (A)P (B). 27
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Experiment 3: 40% of adults aged 40-74 in Rhode Island have pre-diabetes. Two researchers, Dr. Brown and Dr. Green, each indepently sample one person randoml from this population. A= Dr. Brown samples a person with pre-diabetes P (A) = B= Dr. Green samples a person with pre-diabetes P (B) = What is the event A B? P (A B) = What is the event A B? P (A B) = 29
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Example: Suppose we are conducting a hypertension screening program. Assume that the mother and the father are not genetically related and thus their risks of being hypertensive are independent. Suppose in a given family, the probability of the mother being hypertensive is 0.1 and the father being hypertensive is 0.2. What is the probability that both have hypertension? P (A B) = What is the probability that at least one of them has hypertension? P (A B = What is the probability that neither has hypertension? P (Ā B) = 31
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Check independence: Two events A and B are independent if and only if P (A B) = P (A)P (B). Thus, if P (A B) P (A)P (B), A and B are not independent. Recall in Experiment 2, we have Female Male Diabetes 1 3 4 No Diabetes 7 9 16 8 12 20 If one person is randomly selected from these 20, A= this person is female B= this person has diabetes Are A and B independent events? P (A B) = P (female with diabetes) = 1/20 = 0.05 P (A) = P ( female ) = 8/20 =.4, P (B) = P ( Diabetes ) = 4/20 =.2 P (A)P (B) =.4.2 =.08 33
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Do not confuse mutually exclusive and independent events. For mutually exclusive events, P (A B) = P ( ) = 0 For independent events, P (A B) = P (A)P (B) Mutually exclusive: when A happens, B cannot happen Independent events: Whether A happens or not has no effect on B. Unless A or B or both have probability 0, they cannot be both independent and mutually exclusive. 35
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Venn Diagrams 37
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Conditional Probability The probability of an event A, given the occurrence of event B, is defined as P (A B) P (A B) = P (B) If P (B) = 0, the conditional probability is undefined. Another definition of independence: If P (A B) = P (A), events A and B are independent. As long as P (B) 0, this statement is the same as the above definition. The definition of conditional probability also gives the calculation for interceptions. P (A B) = P (A)P (B A) = P (B)P (A B) 39
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Example: The students in a class Male Female Graduate 11 14 25 Undergraduate 5 10 15 16 24 40 If I randomly choose one student from the class, the probability that this student is a graduate student is P (Graduate) = 25/40. Given that the student chosen is female, what is the probability that she is graduate student? Reading from the table, you may do 14/24 directly. P (Graduate F emale) = P (Graduate F emale) P (F emale) = 14/40 24/40 = 14/24 41
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Bayes Theorem P (A j B) = P (A j B) P (B) = P (A j B) n P (B A i )P (A i ) i=1 43
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Example: The prevalence of asthma In a small town in Ohio, 55.6% is female and 44.4% is male. The prevalence of asthma in female is.115 and.106 in male. If we randomly choose one person from this population, what is the probability that this person is female? What if we know this person has asthma? Unconditional probability: P (F ) = 55.6% Conditional probability: P (F A) = P (A F )P (F ) P (A F )P (F ) + P (A M)P (M) =.556.115.556.115 +.444.106 =.576 45
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