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Chapter 3 Probability Probability is the tool that allows the statistician to use sample information to make inferences about or to describe the population from which the sample was drawn. 3.1 Events, Sample Spaces, and Probability Deﬁnition 3.1 Experiment: An experiment is the process by which an observation (or measurement) is obtained. Deﬁnition 3.2 Sample Point: A sample point is the most basic outcome of an experiment. Deﬁnition 3.3 Sample Space: The sample space is the set of all possible outcomes (sample points) of an experiment and is denoted by the symbol S. Example 3.1, page 118: Two coins are tossed and their up faces are recorded. List of all sample points for this experiment. Find the sample space S. See examples for experiments and sample spaces in Table 3.1 on page 119. Extra Example 1: Toss a fair coin and through a die. List of all sample points for this experiment and ﬁnd the sample space S. Venn Diagram: A pictorial method for presenting the sample space where each sample point is represented by a solid dot and labeled accordingly is called the Venn diagram. See Figure 3.2, page 120. Probability: The probability of a sample point is a number between 0 and 1 that measures the likelihood that the outcome will occur when the experiment is performed. Probability Rules for Sample Points: Let pi represent the probability of a sample point i. Then 1. All sample point probabilities must lie between 0 and 1 (0 ≤ pi ≤ 1). 2. The probabilities of all the sample points within a sample space must sum to 1 ( i pi = 1). Example 3.2, page 121. Example 3.3, page 122. 17 Deﬁnition 3.4 Event: An event is a speciﬁc collection of sample points. Simple Event: A single outcome of an experiment is called a simple event. Compound Event: A compound event is a collection of simple events. Probability of an event: The probability of an event A is equal to the sum of the probabilities of the simple events in A. Example 3.4, page 123. Steps for calculating probabilities of events 1. 2. 3. 4. 5. Deﬁne the experiment and the type of observation that will be recorded. List the sample points. Assign probabilities to the sample points. Determine the collection of sample points contained in the event of interest. Sum the sample point probabilities to get the event probability. Extra Example 2: Toss a die and observe the number appearing on the upper face is an experiment. The sample space is S = {1, 2, 3, 4, 5, 6}. We deﬁne the following events Event A = {Observe a odd number}, that is A={1, 3, 5} Event B= {Observe a number less than 4}, that is B={1, 2, 3} Event C = {Observe an even number}, that is C={2, 4, 6} Find the P(A), P(B) and P(C). Combinations Rule: A sample of n elements is to be drawn from a set of N elements. N Then the number of diﬀerent possible samples is denoted by CnN = and is equal to n CnN = N n = N! n!(N − n)! where n! = n(n − 1)(n − 2) . . . 4.3.2.1 (Note 0! = 1) Example 3.7, page 126. Extra Example 3: The personnel director of a company plans to hire two salespeople from a total of four applicants. Suppose she is completely incapable of correctly ranking the applicants according to their ability and in eﬀect, selects them at random. (a) What is the probability that she selects the two best candidates? (b) What is the probability that she selects at least one of the two best candidates? 18 3.2 Unions and Intersections Deﬁnition 3.5 Union of A and B: Let A and B be two events in a sample space S. The union of A and B is the event containing all simple events in A or B or both. We denote the union of A and B by the symbol A ∪ B. Deﬁnition 3.6 Intersection of A and B: Let A and B be two events in a sample space S. The intersection of A and B is the event composed of all simple events that are in both A and B. We denote the intersection of A and B by the symbol A ∩ B or simply AB. Extra Example 4: An experiment can result in 1 of 10 simple events E1 , E2 , . . . E10 which are equally likely; the events A, B, and C are deﬁned as follows Event Simple events A E1 E2 E3 E4 B E3 E4 E5 E6 E7 C E6 E7 E8 (a) List the simple events in the following compound events A∪B, AB, AC, B ∪C, A∪B ∪C and A ∩ B ∩ C. (b) Calculate the probabilities associated with each of the events in part (a) by summing the probabilities of the appropriate simple events. Example 3.9, page 131. 3.3 Complementary Events Deﬁnition 3.7 Complementary event: The complement of an event A, denoted by Ac, consists of all the simple events in the sample space that are not in A. For a complementary event A, P (Ac ) = 1 − P (A). The above equation implies P (Ac ) + P (A) = 1. Example 3.11, page 134. 3.4 The Additive Rule and Mutually Exclusive Events Additive Rule of Probability: The probability of the union of events A and B is the sum of probabilities of events A and B minus the probability of the intersection of events A and B. Given two events A and B, the probability that A or B or both occur is P (A ∪ B) = P (A) + P (B) − P (A ∩ B). 19 Example 3.12, page 135. Deﬁnition 3.8 Mutually Exclusive Event: Two events A and B are said to be mutually exclusive (ME) if when A occurs, B can not occured (and vice versa). That means, two events A and B are said to be mutually exclusive if the event A ∩ B contains no simple event. That is if P (A ∩ B) = 0. Therefore, the events A and Ā are mutually exclusive. If two events A and B are mutually exclusive, then P (A ∪ B) = P (A) + P (B). Example 3.13, page 136. Exercise 3.34, page 138. 3.5 Conditional Probability Sometimes we have additional knowledge/information that might aﬀect the likelihood of the outcome of an experiment or that can be used to better determine the probability of the event of interest. A probability that reﬂects such additional knowledge is called the conditional probability. If A and B are any two events, then the conditional probability of A given B, denoted by P (A|B) is P (A ∩ B) P (A|B) = , P (B) > 0. P (B) If A and B are any two events, then the conditional probability of B given A, denoted by P (B|A) is P (A ∩ B) , P (A) > 0. P (B|A) = P (A) Example 3.15, page 146: Application of Conditional Probability, Statistics by McClave and Sincich Many medical researchers have conducted experiments to examine the relationship between smoking cigarette and lung cancer. Let A represent the event that the individual smokes and let Ac denote the complement of A (the event that the individual does not smoke). Similarly, let B represent the event that the individual develops cancer and let B c be the complement of that event. Then the four sample points associated with the experiment are shown in Table 3.1 and their probabilities for a certain section (a particular population) of the United Sates are provided in Table 3.2. 20 Table 3.1: Sample space for the above example A ∩ B A ∩ Bc Ac ∩ B Ac ∩ B c Table 3.2: Probabilities of smoking and Develops Cancer Smoker Yes, B No, B c Yes, A 0.05 0.20 No, Ac 0.03 0.72 Total 0.08 0.92 developing cancer Total 0.25 0.75 1.00 Suppose an individual has randomly selected from this population. (a) What is the probability that the randomly selected individual is a smoker? Ans: P (A) = P (A ∩ B) + P (A ∩ B c ) = 0.05 + 0.20 = 0.25 (b) What is the probability that a randomly selected individual develops cancer? Ans: P (B) = P (A ∩ B) + P (Ac ∩ B) = 0.05 + 0.03 = 0.08 (c) What is the probability that a randomly selected smoker develops cancer? OR What is the probability that a randomly selected individual develops cancer given that he/she smokes cigarette? OR If you know an individual smoker in that area, what is the probability that he/she develops cancer? Ans: P (B|A) = 0.05 P (A ∩ B) = = 0.20 P (A) 0.25 (d) What is the probability that a randomly selected smoker does not develop cancer? OR What is the probability that a randomly selected individual does not develop cancer given that he/she smokes cigarette? OR If you know an individual smoker in that area, what is the probability that he/she does not develop cancer? Ans: P (B c |A) = P (A ∩ B c ) 0.20 = = 0.80 P (A) 0.25 21 (e) What is the probability that a randomly selected individual develops cancer given that he/she does not smoke cigarette? Ans: P (Ac ∩ B) 0.03 P (B|Ac) = = = 0.04 c P (A ) 0.75 (f ) What is the probability that a randomly selected individual does not develop cancer given that he/she does not smoke cigarette? Ans: P (B c |Ac ) = P (Ac ∩ B c ) 0.72 = = 0.96 c P (A ) 0.75 The conditional probability that an individual smoker develops cancer (0.20) is ﬁve times the probability that a nonsmoker develops cancer (0.04). This does not necessarily imply that smoking causes cancer, but it does suggest a link between smoking and cancer. 3.6 The Multiplicative Rule and Independent Events Multiplicative Rule of Probability: The probability that both A and B occur is P (A ∩ B) = P (AB) = P (A|B)P (B) = P (B|A)P (A). (3.1) Example 3.17, page 145. Deﬁnition 3.9 Independent Events: Two events A and B are said to be independent if any one of the followings holds: (i) P (A|B) = P (A) (ii) P (B|A) = P (B) (iii) P (A ∩ B) = P (A)P (B). Otherwise, the events are said to be dependent. Example 3.19, page 147. Note: Mutually exclusive does not necessarily mean independent. For example, the events A and C in extra example 2 are mutually exclusive but they are not independent. Probability of Intersection of Two Independent Events If events A and B are independent, then P (A ∩ B) = P (A)P (B) The converse is also true: If P (A)P (B) = P (A ∩ B), then events A and B are independent. Exercise 3.50, page 151. 22 3.7 Random Sampling Deﬁnition 3.10 Simple Random Sample: Let N and n represent the number of elements in the population and sample respectively. If the sampling is conducted in such a way that ! samples has an equal probability of being selected, the sampling each of the CnN = n!(NN−n)! is said to be random and the resulting sample is said to be a simple random sample. Here N! CnN = , n!(N − n)! where n! = n(n − 1)(n − 2)......4.3.2.1. Then 5! = 5.4.3.2.1 = 120. Example 3.22, page 155. 23