Slide 1 Statistics Workshop Tutorial 4 Probability Probability Distributions

Slide 2 Created by Tom Wegleitner, Centreville, Virginia Probability

Slide 3 Copyright © 2004 Pearson Education, Inc. Definitions  Event Any collection of results or outcomes of a procedure.  Simple Event An outcome or an event that cannot be further broken down into simpler components.  Sample Space Consists of all possible simple events. That is, the sample space consists of all outcomes that cannot be broken down any further.

Slide 4 Copyright © 2004 Pearson Education, Inc. Notation for Probabilities P - denotes a probability. A, B, and C - denote specific events. P (A) - denotes the probability of event A occurring.

Slide 5 Copyright © 2004 Pearson Education, Inc. Basic Rules for Computing Probability Rule 1: Relative Frequency Approximation of Probability Conduct (or observe) a procedure a large number of times, and count the number of times event A actually occurs. Based on these actual results, P(A) is estimated as follows: P(A) =P(A) = number of times A occurred number of times trial was repeated

Slide 6 Copyright © 2004 Pearson Education, Inc. Basic Rules for Computing Probability Rule 2: Classical Approach to Probability (Requires Equally Likely Outcomes) Assume that a given procedure has n different simple events and that each of those simple events has an equal chance of occurring. If event A can occur in s of these n ways, then P(A) = number of ways A can occur number of different simple events s n =

Slide 7 Copyright © 2004 Pearson Education, Inc. Law of Large Numbers As a procedure is repeated again and again, the relative frequency probability (from rule 1) of an event tends to approach the actual probability.

A simple example of randomness involves a coin toss. The outcome of the toss is uncertain. Since the coin tossing experiment is unpredictable, the outcome is said to exhibit randomness. Even though individual flips of a coin are unpredictable, if we flip the coin a large number of times, a pattern will emerge. Roughly half of the flips will be heads and half will be tails.

Slide 9 Copyright © 2004 Pearson Education, Inc. Illustration of Law of Large Numbers

Slide 10 Copyright © 2004 Pearson Education, Inc. Probability Limits  The probability of an event that is certain to occur is 1.  The probability of an impossible event is 0.  0  P(A)  1 for any event A.

Slide 11 Copyright © 2004 Pearson Education, Inc. Possible Values for Probabilities Figure 3-2

Slide 13 Copyright © 2004 Pearson Education, Inc. Created by Tom Wegleitner, Centreville, Virginia Probability Distributions

Slide 14 Copyright © 2004 Pearson Education, Inc. Overview This chapter will deal with the construction of probability distributions by combining the methods of descriptive statistics presented in Chapter 2 and those of probability presented in Chapter 3. Probability Distributions will describe what will probably happen instead of what actually did happen.

Slide 15 Copyright © 2004 Pearson Education, Inc. Figure 4-1 Combining Descriptive Methods and Probabilities In this chapter we will construct probability distributions by presenting possible outcomes along with the relative frequencies we expect.

Slide 16 Copyright © 2004 Pearson Education, Inc. Definitions  A random variable is a variable (typically represented by x ) that has a single numerical value, determined by chance, for each outcome of a procedure.  A probability distribution is a graph, table, or formula that gives the probability for each value of the random variable.

Slide 17 Copyright © 2004 Pearson Education, Inc. Definitions  A discrete random variable has 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.  A continuous random variable has infinitely many values, and those values can be associated with measurements on a continuous scale in such a way that there are no gaps or interruptions.

Slide 18 Copyright © 2004 Pearson Education, Inc. Graphs The probability histogram is very similar to a relative frequency histogram, but the vertical scale shows probabilities. Figure 4-3

Slide 19 Copyright © 2004 Pearson Education, Inc. Requirements for Probability Distribution P ( x ) = 1 where x assumes all possible values  0  P ( x )  1 for every individual value of x

Slide 20 Copyright © 2004 Pearson Education, Inc. Mean, Variance and Standard Deviation of a Probability Distribution µ =  [x P(x)] Mean  2 =  [ (x – µ) 2 P(x )] Variance  2 =  [ x 2 P ( x )] – µ 2 Variance (shortcut )  =  [ x 2 P ( x )] – µ 2 Standard Deviation

Slide 21 Copyright © 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 = μ + 2σ Minimum usual value = μ – 2σ

Slide 22 Copyright © 2004 Pearson Education, Inc. Identifying Unusual Results With Probabilities Rare Event Rule If, under a given assumption (such as the assumption that boys and girls are equally likely), the probability of a particular observed event (such as 13 girls in 14 births) is extremely small, we conclude that the assumption is probably not correct.  Unusually high: x successes among n trials is an unusually high number of successes if P(x or more) is very small (such as 0.05 or less).  Unusually low: x successes among n trials is an unusually low number of successes if P(x or fewer) is very small (such as 0.05 or less).

Slide 23 Now we are ready for Part 14 of Day 1