Mathematics
Conditional Probability
Conditional probability is the likelihood of an event occurring given that another event has already occurred. It is calculated by dividing the probability of the intersection of the two events by the probability of the given event. This concept is fundamental in understanding the relationship between events and is widely used in fields such as statistics, machine learning, and decision-making.
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11 Key excerpts on "Conditional Probability"
No longer available |Learn more- Jessica Utts, Robert Heckard(Authors)
- 2015(Publication Date)
- Cengage Learning EMEA(Publisher)
EXAMPLE 7.11 The Probability That Alicia Has to Answer a Question In Case Study 7.1 (p. 222), Alicia was worried that her statistics teacher might call on her to answer a question during class. Define Event A Alicia is selected to answer question 1. Event B Alicia is selected to answer question 2. In Example 7.3, we determined that P (A) 1/50. Are A and B independent? Note that they refer to different random circumstances, but the probability that B occurs is affected by knowing whether A occurred. If Alicia answered question 1, her name is already out of the bag, so the probability that B occurs is 0. If event A does not occur, there are 49 names left in the bag including Alicia’s, so the probability that B occurs is 1/49. Therefore, knowing whether A occurred changes the probability that B occurs. Conditional Probabilities Suppose that two events are dependent. From the definition of dependent events, it fol-lows that knowing that one event will occur (or has occurred) does change the probability that the other occurs. Therefore, we need to distinguish between two probabilities: • P (B), the unConditional Probability that the event B occurs • P (B|A), read “probability of B given A,” the Conditional Probability that the event B occurs given that we know A has occurred or will occur DEFINITION The Conditional Probability of the event B, given that the event A has occurred or will occur, is the long-run relative frequency with which event B occurs when circumstances are such that A has occurred or will occur. This probability is written as P (B|A) . EXAMPLE 7.12 Probability That a Teenager Gambles Differs for Boys and Girls Based on a sur-vey of nearly all of the ninth- and twelfth-graders in Minnesota public schools in 1998, a total of 78,564 students, Stinchfield (2001) reported proportions admitting that they had gambled at least once a week during the previous year.
eBook - PDFEntropy Demystified: The Second Law Reduced To Plain Common Sense
The Second Law Reduced to Plain Common Sense
- Arieh Ben-naim(Author)
- 2007(Publication Date)
- World Scientific(Publisher)
Both of these can either be condi-tional or unconditional. In this book, we shall use only scien-tific, hence, objective probabilities. In using probability in all the sciences, we always assume that the probabilities are given either explicitly or implicitly by a given recipe on how to calcu-late these probabilities. While these are sometimes very easy to calculate, at other times, they are very difficult, 21 but you can 21 For instance, the probability of drawing three red marbles, five blue marbles and two green marbles from an urn containing 300 marbles, 100 of each color, given that the marbles are identical and that you drew 10 marbles at random, and so on . . . This is a slightly more difficult problem, and you might not be able to calculate it, but the probability of this event is “there” in the event itself. Similarly, the probability of finding two atoms at a certain distance from each other, in a liquid at a given temper-ature and pressure, given that you know and accept the rules of statistical mechanics, and that you know that these rules have been extremely useful in predicting many average properties of macroscopic properties, etc. This probability is objective! You might not be able to calculate it, but you know it is “there” in the event. 46 Entropy Demystified always assume that they are “there” in the event, as much as mass is attached to any piece of matter. 2.4.2. Conditional Probability and cause and effect The “condition” in the Conditional Probability of an event may or may not be the cause of the event. Consider the following two examples: 1. The Conditional Probability that the patient will die of lung cancer, given that he or she is a heavy smoker, is 9/10. 2. The Conditional Probability that the patient is a heavy smoker given that he or she has lung cancer, is 9/10. Clearly, the information given in the first condition is the cause (or the very probable cause) of the occurrence of lung cancer.
eBook - PDF- Prem S. Mann(Author)
- 2016(Publication Date)
- Wiley(Publisher)
It is obvious from the formula for joint probability that if we know the probability of an event A and the joint probability of events A and B, then we can calculate the Conditional Probability of B given A. Calculating Conditional Probability If A and B are two events, then, P( B ∣ A) = P( A and B) P( A) and P( A ∣ B) = P( A and B) P( B) given that P( A) ≠ 0 and P( B) ≠ 0. EXAMPLE 4–24 Major and Status of a Student The probability that a randomly selected student from a college is a senior is .20, and the joint probability that the student is a computer science major and a senior is .03. Find the Conditional Probability that a student selected at random is a computer science major given that the student is a senior. Solution From the given information, the probability that a randomly selected student is a senior is: P( senior ) = .20 The probability that the selected student is a senior and computer science major is: P( senior and computer science major ) = .03 Hence, the Conditional Probability that a randomly selected student is a computer science major given that he or she is a senior is: P( computer science major ∣ senior ) = P( senior and computer science major ) ∕ P( senior ) = .03 .20 = .15 Thus, the (conditional) probability is .15 that a student selected at random is a computer science major given that he or she is a senior. ◼ Calculating the Conditional Probability of an event. 4.4.4 Joint Probability of Mutually Exclusive Events We know from an earlier discussion that two mutually exclusive events cannot happen together. Consequently, their joint probability is zero. 4.4 Intersection of Events and the Multiplication Rule 155 EXAMPLE 4–25 Car Loan Application Consider the following two events for an application filed by a person to obtain a car loan: A = event that the loan application is approved R = event that the loan application is rejected What is the joint probability of A and R? Solution The two events A and R are mutually exclusive.
eBook - ePub- Dov M. Gabbay, Paul Thagard, John Woods(Authors)
- 2011(Publication Date)
- North Holland(Publisher)
(For continuous random variables, we replace the p (x i) by values given by a density function, and replace the sum by an integral.) A conditional expectation is an expectation of a random variable with respect to a Conditional Probability distribution. Let X and Y be two random variables with joint distribution The conditional expectation E (Y | X = x j) of Y given X = x j is given by: (Again, this has a continuous version.) We may generalize to conditional expectations involving more random variables. So the importance of Conditional Probability in probability theory is beyond dispute. The same can be said of its role in many philosophical applications. 3. Philosophical Applications Conditional Probability is near-ubiquitous in both the methodology — in particular, the use of statistics and game theory — of the sciences and social sciences, and in their specific theories. Various central concepts in statistics are defined in terms of conditional probabilities: significance level, power, sufficient statistics, ancillarity, maximum likelihood estimation, Fisher information, and so on. Game theorists appeal to conditional probabilities for calculating the expected payoffs in correlated equilibrium; computing the Bayesian equilibrium in games of incomplete information; in certain Bayesian dynamic updating models of equilibrium selection in repeated games; and so on. Born's rule in quantum mechanics is often understood as a method of calculating the Conditional Probability of a particular measurement outcome, given that a measurement of a certain kind is performed on a system in a certain state. In medical and clinical psychological testing, conditional probabilities of the form P (disorder | positive test) (“diagnosticity”) and P (positive test | disorder) (“sensitivity”) take centre stage
eBook - PDFStatistics
Principles and Methods
- Richard A. Johnson, Gouri K. Bhattacharyya(Authors)
- 2019(Publication Date)
- Wiley(Publisher)
4 Probability 1 Probability of an Event 2 Methods of Assigning Probability 3 Event Operations and Two Laws of Probability 4 Conditional Probability and Independence 5 Bayes’ Theorem 6 Random Sampling from a Finite Population Uncertainty of Weather Forecasts Today’s forecast: Increasing cloudiness with a 25% chance of snow. Huntstock/Stockbyte/Getty Images Inc. Probabilities express the chance of events that cannot be predicted with certainty. Even unlikely events sometimes occur. 92 CHAPTER 4/PROBABILITY In Chapter 1, we introduced the notions of sample and statistical population in the context of investigations where the outcomes exhibit variation. Although complete knowledge of the statisti- cal population remains the target of an investigation, we typically only have the partial information contained in a sample. Chapter 2 focused on some methods for describing the salient features of a data set by graphical presentations and calculation of the mean, standard deviation, and other summary statistics. However, when the data set represents a sample from a statistical population, its description is only a preliminary part of a statistical analysis. Our major goal is to make gener- alizations or inferences about the target population on the basis of information obtained from the sample data. It is the subject of probability that is essential for understanding the reasoning that leads to such generalizations. In everyday conversations, we all use expressions of the kind: “Most likely our team will win this Saturday.” “It is unlikely that the weekend will be cold.” “I have a 50–50 chance of getting a summer job at the camp.” The phrases “most likely,” “probable,” “quite likely,” and so on are used qualitatively to indicate the chance that an event will occur. Probability, as a subject, provides a means of quantifying uncertainty. In general terms, the probability of an event is a numerical value that gauges how likely it is that the event will occur.
eBook - PDFProbability Theory
A First Course in Probability Theory and Statistics
- Werner Linde(Author)
- 2016(Publication Date)
- De Gruyter(Publisher)
2 Conditional Probabilities and Independence 2.1 Conditional Probabilities In order to motivate the definition of conditional probabilities, let us start with the following easy example. Example 2.1.1. Roll a fair die twice. The probability of the event “sum of both rolls equals 5” is 1/9. Suppose now we were told that the first roll was an even number. Does this additional information make the event “sum equals 5” more likely? Or does it even diminish the probability of its occurrence? To answer this question, we apply the so-called technique of “restricting the sample space.” Since we know that the event B = {First roll is even} had occurred, we may rule out elements in B c and re- strict our sample space. Choose B as new sample space. Its cardinality is 18. Moreover, under this condition, an event A occurs if and only if A ∩ B does so. Hence, the “new” probability of A under condition B, written P(A|B), is given by P(A|B) = #(A ∩ B) #(B) = #(A ∩ B) 18 . (2.1) In the question above, we asked for P(A|B), where A = {Sum of both rolls equals 5} = {(1, 4), (2, 3), (3, 2), (4, 1)} . Since A ∩ B = {(2, 3), (4, 1)}, we obtain P(A|B) = 2/18 = 1/9. Consequently, in this case, condition B does not change the probability of the occurrence of A. Define now A as a set of pairs adding to 6. Then P(A) = 5/36, while the condi- tional probability remains 1/9. Note that now A ∩ B = {(2, 4), (4, 2)}. Thus, in this case, condition B makes the occurrence of A less likely. Before we state the definition of conditional probabilities in the general case, let us rewrite eq. (2.1) as follows: P(A|B) = #(A ∩ B) #(B) = #(A ∩ B)/36 #(B)/36 = P(A ∩ B) P(B) . (2.2) Equation (2.2) gives us a hint to introduce conditional probabilities in the general setting. Definition 2.1.2. Let (K, A, P) be a probability space. Given events A, B ∈ A with P(B) > 0, the probability of A under condition B is defined by P(A|B) = P(A ∩ B) P(B) . (2.3)
- David Anderson, Dennis Sweeney, Thomas Williams, Jeffrey Camm(Authors)
- 2020(Publication Date)
- Cengage Learning EMEA(Publisher)
Conditional Probability P s A | B d 5 P s A > B d P s B d (4.7) or P s B | A d 5 P s A > B d P s A d (4.8) The Venn diagram in Figure 4.8 is helpful in obtaining an intuitive understanding of Conditional Probability. The circle on the right shows that event B has occurred; the portion of the circle that overlaps with event A denotes the event ( A > B ). We know that once event B has occurred, the only way that we can also observe event A is for the event ( A > B ) to occur. Thus, the ratio P ( A > B )/ P ( B ) provides the Conditional Probability that we will observe event A given that event B has already occurred. Let us return to the issue of discrimination against the female officers. The marginal probability in row 1 of Table 4.5 shows that the probability of promotion of an officer is P ( A ) 5 .27 (regardless of whether that officer is male or female). However, the critical issue in the discrimination case involves the two conditional probabilities P ( A | M ) and P ( A | W ). That is, what is the probability of a promotion given that the officer is a man, and what is the probability of a promotion given that the officer is a woman? If these two Event A > B Event B Event A FIGURE 4.8 Conditional Probability P ( A ∣ B ) = P ( A ∩ B )/ P ( B ) Copyright 2020 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 196 Chapter 4 Introduction to Probability probabilities are equal, a discrimination argument has no basis because the chances of a promotion are the same for male and female officers.
eBook - PDF- Vladimir I. Rotar(Author)
- 2012(Publication Date)
- Chapman and Hall/CRC(Publisher)
The symbol P ( A | B ) is read as “the probability of A given B .” From the definition of independence (1.1.1), it follows that If A and B are independent, then P ( A | B ) = P ( A ) . (2.1.2) Indeed, if P ( AB ) = P ( A ) P ( B ) , then the factor P ( B ) in (2.1.1) cancels. EXAMPLE 1. Suppose 4% of students have A’s for all mathematics courses, and 10% for a probability course. Mary received an A for a probability course. What is the probability that she has A’s for all mathematics courses? Keeping in mind that the event { A for Mathematics } ⊂ { A for Probability } , we have P ( A for Mathematics | A for Probability ) = P ( A for Mathematics AND A for Probability ) P ( A for Probability ) = P ( A for Mathematics ) P ( A for Probability ) = 0 . 04 0 . 1 = 0 . 4 . 38 2. INDEPENDENCE AND Conditional Probability EXAMPLE 2 is well known. You visit a family with two children, and you know that one child is a boy. For simplicity, consider the case of equally likely outcomes in the sample space { bb , bg , gb , gg } . What is the probability that the other child is also a boy? 1 Often, people give the answer 1 / 2, reasoning that it is equally likely that the other kid is a boy or girl. The correct answer is 1 / 3, and apparently the mistake consists in unconscious identifying the outcomes bg and gb . To adjust our intuition, observe that since it is given that one child is a boy, we should consider only the outcomes bb , bg , gb . They are equally likely, and hence, the probability of interest is 1 / 3. The formal proof should run as follows. Let events A = { bb } , B = { bb , bg , gb } . Since A ⊂ B , P ( A | B ) = P ( AB ) P ( B ) = P ( A ) P ( B ) = 1 / 4 3 / 4 = 1 3 . EXAMPLE 3. Each day, the president of a company inspects one of the company’s five divisions. The president chooses a division at random, and the choice does not depend on which divisions were inspected on previous days. Suppose a division has not been in-spected for 30 consecutive days.
eBook - PDF- Henk Tijms(Author)
- 2012(Publication Date)
- Cambridge University Press(Publisher)
. . , B n . A nice illustration of the law of conditional prob- ability is provided by the craps example in Section 3.3 of Chapter 3. Another nice illustrative example is the following one. Example 8.6 The upcoming Tour de France bicycle tournament will take place from July 1 through July 23. One hundred eighty cyclists will participate in the 8.2 Law of Conditional Probability 267 event. What is the probability that two or more participating cyclists will have birthdays on the same day during the tournament? Solution. Denoting by A the event that two or more participating cyclists will have birthdays on the same day during the tournament, event A can occur only if one of the mutually exclusive events B 2 , . . . , B 180 occurs. Event B i occurs when exactly i participating cyclists have their birthdays during the tournament. The Conditional Probability P (A | B i ) is easy to calculate. It refers to the birthday problem with i persons coming from a “planet” where the year has 23 days. Using the standard trick of calculating the complementary probability that each person has a different birthday, we find P (A | B i ) = 1 − 23×22×···×(23−i +1) 23 i , 2 ≤ i ≤ 23 1, i ≥ 24. Obviously, P (A | B 0 ) = P (A | B 1 ) = 0. The probability P (B i ) is given by P (B i ) = 180 i 23 365 i 1 − 23 365 180−i , 0 ≤ i ≤ 180. Putting the pieces together, we obtain from P (A) = ∑ 180 i =2 P (A | B i )P (B i ) that P (A) = 1 − P (B 0 ) − P (B 1 ) − 23 i =2 23 × 22 × · · · × (23 − i + 1) 23 i P (B i ). This yields the value 0.8841 for the probability P (A). The following example uses conditional probabilities to solve a real-life problem. Example 8.7 Two candidates A and B remain in the final of a television game show. At this point, each candidate must spin a wheel of fortune. The numbers 1, 2, . . . , 100 are listed on the wheel and when the wheel has stopped spinning, a pointer randomly stops on one of the numbers.
eBook - PDF- Jeffrey Camm, James Cochran, Michael Fry, Jeffrey Ohlmann(Authors)
- 2020(Publication Date)
- Cengage Learning EMEA(Publisher)
Probability: An Introduction to Modeling Uncertainty C O N T E N T S ANALYTICS IN ACTION: NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 4.1 EVENTS AND PROBABILITIES 4.2 SOME BASIC RELATIONSHIPS OF PROBABILITY Complement of an Event Addition Law 4.3 Conditional Probability Independent Events Multiplication Law Bayes’ Theorem 4.4 RANDOM VARIABLES Discrete Random Variables Continuous Random Variables 4.5 DISCRETE PROBABILITY DISTRIBUTIONS Custom Discrete Probability Distribution Expected Value and Variance Discrete Uniform Probability Distribution Binomial Probability Distribution Poisson Probability Distribution 4.6 CONTINUOUS PROBABILITY DISTRIBUTIONS Uniform Probability Distribution Triangular Probability Distribution Normal Probability Distribution Exponential Probability Distribution SUMMARY 198 GLOSSARY 198 PROBLEMS 200 AVAILABLE IN THE MINDTAP READER: APPENDIX: DISCRETE PROBABILITY DISTRIBUTIONS WITH R APPENDIX: CONTINUOUS PROBABILITY DISTRIBUTIONS WITH R Chapter 4 Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 158 Chapter 4 Probability: An Introduction to Modeling Uncertainty Uncertainty is an ever-present fact of life for decision makers, and much time and effort are spent trying to plan for, and respond to, uncertainty. Consider the CEO who has to make decisions about marketing budgets and production amounts using forecasted demands. Or consider the financial analyst who must determine how to build a client’s portfolio of stocks and bonds when the rates of return for these investments are not known with certainty.
eBook - PDFBoole's Logic and Probability
A Critical Exposition from the Standpoint of Contemporary Algebra, Logic and Probability Theory
- T. Hailperin(Author)
- 1986(Publication Date)
- North Holland(Publisher)
. . and a second case as follows : Case 11.-When some of the events are conditioned. If there be given the probability p that if the event X occur, the event Y will occur, and if the probability of the antecedent X 234 PROBABILITY FROM BOOLE’S VIEWPOINT be not given, resolve the proposition into the two following, viz: Probability of X = c, Probability of X Y = cp. If the quaesitum be the probability that if the event W occur, the event Z will occur, determine separately, by the previous case, the terms of the fraction Prob. W Z Prob. W ’ and the fraction itself will express the probability sought. It is understood in this case that X , X W, Z may be any compound events whatsoever. The expressions X Y and W Z represent the products of the symbolical expressions of X and Y and W and 2, formed according to the rule of the Calculus of Logic. It thus appears that when conditioned events are part of the data then new parameters, e.g. the c = Probability of X in Case I1 just quoted, can enter. But then Boole’s general method isn’t quite what one may have understood it to be! For when he says in Proposition IV “Giren the probabilities of any system of events ; to determine by u general method the consequent or derired probubility of any other erent” one would naturally suppose it to mean that the derived probability would be in terms of the given probabilities.
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