Slide 1 Statistics Workshop Tutorial 4 Probability Probability Distributions

Slide 1

Statistics Workshop Tutorial 4

• Probability• Probability Distributions

Slide 2

Created by Tom Wegleitner, Centreville, Virginia

Probability

Copyright © 2004 Pearson Education, Inc.

Slide 3

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.

Copyright © 2004 Pearson Education, Inc.

Slide 4Notation for Probabilities

P - denotes a probability.

A, B, and C - denote specific events.

P (A) - denotes the probability of event A occurring.

Copyright © 2004 Pearson Education, Inc.

Slide 5Basic Rules for

Computing ProbabilityRule 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) = number of times A occurred

number of times trial was repeated

Copyright © 2004 Pearson Education, Inc.

Slide 6Basic Rules for

Computing ProbabilityRule 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

sn =

Copyright © 2004 Pearson Education, Inc.

Slide 7Law 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.

Copyright © 2004 Pearson Education, Inc.

Slide 9Illustration of Law of Large Numbers

Copyright © 2004 Pearson Education, Inc.

Slide 10Probability 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.

Copyright © 2004 Pearson Education, Inc.

Slide 11Possible Values for

Probabilities

Figure 3-2

Copyright © 2004 Pearson Education, Inc.

Slide 13

Created by Tom Wegleitner, Centreville, Virginia

Probability Distributions

Copyright © 2004 Pearson Education, Inc.

Slide 14OverviewThis 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.

Copyright © 2004 Pearson Education, Inc.

Slide 15

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.

Copyright © 2004 Pearson Education, Inc.

Slide 16Definitions

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.

Copyright © 2004 Pearson Education, Inc.

Slide 17Definitions

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.

Copyright © 2004 Pearson Education, Inc.

Slide 18GraphsThe probability histogram is very similar to a relative frequency histogram, but the vertical scale shows probabilities.

Figure 4-3

Copyright © 2004 Pearson Education, Inc.

Slide 19Requirements for

Probability Distribution

P(x) = 1 where x assumes all possible values

0 P(x) 1 for every individual value of x

Copyright © 2004 Pearson Education, Inc.

Slide 20Mean, Variance and

Standard Deviation of a Probability Distribution

µ = [x • P(x)] Mean

2 = [(x – µ)2 • P(x)] Variance

2 = [x2

• P(x)] – µ 2 Variance (shortcut)

= [x 2 • P(x)] – µ 2 Standard Deviation

Copyright © 2004 Pearson Education, Inc.

Slide 21Identifying 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

Copyright © 2004 Pearson Education, Inc.

Slide 22Identifying Unusual Results

With ProbabilitiesRare 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