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
Top Related