1. 4.1 - 1 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Lecture Slides Elementary Statistics Eleventh Edition and the Triola Statistics Series by Mario F. Triola

2. 4.1 - 2 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Chapter 4 Probability 4-1 Review and Preview 4-2 Basic Concepts of Probability 4-3 Addition Rule 4-4 Multiplication Rule: Basics 4-5 Multiplication Rule: Complements and Conditional Probability 4-6 Probabilities Through Simulations 4-7 Counting

3. 4.1 - 3 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Section 4-5 Multiplication Rule: Complements and Conditional Probability

4. 4.1 - 4 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Key Concepts Probability of “at least one”: Find the probability that among several trials, we get at least one of some specified event. Conditional probability: Find the probability of an event when we have additional information that some other event has already occurred.

5. 4.1 - 5 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Complements: The Probability of “At Least One”  The complement of getting at least one item of a particular type is that you get no items of that type.  “At least one” is equivalent to “one or more.”

6. 4.1 - 6 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Finding the Probability of “At Least One” To find the probability of at least one of something, calculate the probability of none, then subtract that result from 1. That is, P(at least one) = 1 – P(none).

7. 4.1 - 7 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Conditional Probability A conditional probability of an event is a probability obtained with the additional information that some other event has already occurred. denotes the conditional probability of event B occurring, given that event A has already occurred, and it can be found by dividing the probability of events A and B both occurring by the probability of event A:

8. 4.1 - 8 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Intuitive Approach to Conditional Probability The conditional probability of B given A can be found by assuming that event A has occurred, and then calculating the probability that event B will occur.

9. 4.1 - 9 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Confusion of the Inverse To incorrectly believe that and are the same, or to incorrectly use one value for the other, is often called confusion of the inverse.

10. 4.1 - 10 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Recap In this section we have discussed:  Concept of “at least one.”  Conditional probability.  Intuitive approach to conditional probability.

11. 4.1 - 11 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Chapter 5 Probability Distributions 5-1 Review and Preview 5-2 Random Variables 5-3 Binomial Probability Distributions 5-4 Mean, Variance and Standard Deviation for the Binomial Distribution 5-5 Poisson Probability Distributions

12. 4.1 - 12 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Section 5-1 Review and Preview

13. 4.1 - 13 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Review and Preview This chapter combines the methods of descriptive statistics presented in Chapter 2 and 3 and those of probability presented in Chapter 4 to describe and analyze probability distributions. Probability Distributions describe what will probably happen instead of what actually did happen, and they are often given in the format of a graph, table, or formula.

14. 4.1 - 14 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Preview In order to fully understand probability distributions, we must first understand the concept of a random variable, and be able to distinguish between discrete and continuous random variables. In this chapter we focus on discrete probability distributions. In particular, we discuss binomial and Poisson probability distributions.

15. 4.1 - 15 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Combining Descriptive Methods and Probabilities In this chapter we will construct probability distributions by presenting possible outcomes along with the relative frequencies we expect.

16. 4.1 - 16 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Section 5-2 Random Variables

17. 4.1 - 17 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Key Concept This section introduces the important concept of a probability distribution, which gives the probability for each value of a variable that is determined by chance. Give consideration to distinguishing between outcomes that are likely to occur by chance and outcomes that are “unusual” in the sense they are not likely to occur by chance.

18. 4.1 - 18 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Key Concept The concept of random variables and how they relate to probability distributions Distinguish between discrete random variables and continuous random variables Develop formulas for finding the mean, variance, and standard deviation for a probability distribution Determine whether outcomes are likely to occur by chance or they are unusual (in the sense that they are not likely to occur by chance)

19. 4.1 - 19 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Random Variable Probability Distribution  Random variable a variable (typically represented by x) that has a single numerical value, determined by chance, for each outcome of a procedure  Probability distribution a description that gives the probability for each value of the random variable; often expressed in the format of a graph, table, or formula

20. 4.1 - 20 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Discrete and Continuous Random Variables  Discrete random variable either a finite number of values or countable number of values, where “countable” refers to the fact that there might be infinitely many values, but they result from a counting process  Continuous random variable infinitely many values, and those values can be associated with measurements on a continuous scale without gaps or interruptions

21. 4.1 - 21 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Graphs The probability histogram is very similar to a relative frequency histogram, but the vertical scale shows probabilities.

22. 4.1 - 22 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Requirements for Probability Distribution where x assumes all possible values. for every individual value of x.

23. 4.1 - 23 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Mean Variance Variance (shortcut ) Standard Deviation Mean, Variance and Standard Deviation of a Probability Distribution

24. 4.1 - 24 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Round results by carrying one more decimal place than the number of decimal places used for the random variable. If the values of are integers, round,  and to one decimal place. Roundoff Rule for,, and

25. 4.1 - 25 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Identifying Unusual Results Range Rule of Thumb According to the range rule of thumb, most values should lie within 2 standard deviations of the mean. We can therefore identify “unusual” values by determining if they lie outside these limits: Maximum usual value = Minimum usual value =

26. 4.1 - 26 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Identifying Unusual Results Probabilities Rare Event Rule for Inferential Statistics If, under a given assumption (such as the assumption that a coin is fair), the probability of a particular observed event (such as 992 heads in 1000 tosses of a coin) is extremely small, we conclude that the assumption is probably not correct.

27. 4.1 - 27 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Identifying Unusual Results Probabilities Using Probabilities to Determine When Results Are Unusual  Unusually high: x successes among n trials is an unusually high number of successes if.  Unusually low: x successes among n trials is an unusually low number of successes if.

28. 4.1 - 28 Copyright © 2010, 2007, 2004 Pearson Education, Inc. The expected value of a discrete random variable is denoted by E, and it represents the mean value of the outcomes. It is obtained by finding the value of Expected Value

29. 4.1 - 29 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Recap In this section we have discussed:  Combining methods of descriptive statistics with probability.  Probability histograms.  Requirements for a probability distribution.  Mean, variance and standard deviation of a probability distribution.  Random variables and probability distributions.  Identifying unusual results.  Expected value.