© 2005 McGraw-Hill Ryerson Ltd. 5-1

StatisticsA First Course

Donald H. SandersRobert K. Smidt

Aminmohamed AdatiaGlenn A. Larson

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Chapter 5

Probability Distributions

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Chapter 5 - Topics

• Binomial Experiments• Determining Binomial Probabilities• The Poisson Distribution• The Normal Distribution• Normal Approximation of the Binomial

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Binomial Experiments

• Properties of a Binomial Experiment– Same action (trial) is repeated a fixed

number of times– Each trial is independent of the others– Two possible outcomes – success or failure– Constant probability of success for each trial

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Determining Binomial Probabilities

• Combinations– Selection of r items from a set of n distinct

objects without regard to the order in which r items are picked

Combination Rule

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Determining Binomial Probabilities

• Binomial Probability– Probability of correctly guessing exactly r items

from a set of n distinct objects without regard to the order in which r items are picked

Binomial Probability Formula

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Our QuickQuiz probability distribution.

Figure 5.1 (including table)

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Variance of Binomial Distribution Formula

Standard Deviation of Binomial Distribution Formula

Expected Value (Mean) of Binomial Distribution Formula

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The Poisson Distribution

• Discrete probability distribution• Used to determine the number of specified

occurrences that take place within a unit of time, distance, area, or volume

Poisson Distribution Formula

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The Normal Distribution

• Continuous probability distribution• Used to investigate the probability that the

variable assumes any value within a given interval of values

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Normal Distribution.

Figure 5.4

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Probability of breaking strength between 110 and 120.

Figure 5.5

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Both intervals extend from the mean (z = 0) to 1 standard

deviation above themean (z = 1.00).

Figure 5.6

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The probability that a z value selected at random will fall between

0 and 2.27 or between–2.27 and 0 is .4884.

Figure 5.7

Calculating Probabilities for the Standard Normal Distribution

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The area under the normal curve between vertical lines drawn at

z = –1.73 and z = +2.45 is .9511.

Figure 5.8

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The area under the normal curve between a z value of –1.54 and

a z value of –.76 is .1618.

Figure 5.9

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The area under the normal curve to the left of a z value of

–1.96 is .0250.

Figure 5.10

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The area under the normal curve to the left of a z value

of 1.42 is .9222.

Figure 5.11

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The Normal Distribution• Computing Probabilities for Any Normally Distributed

Variable– z scores correspond to the number of standard deviations a

data value is from the mean– Any value can be converted to a standard score (z score)

Convert x value to z score formula

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The z score interval corresponding to 70 < x < 130

Figure 5.13

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The Normal Distribution

• Finding Cut-off Scores for Normally Distributed Variables – Given the area under the standard normal curve, the z

score method can be used to calculate the cut off point

Convert z score to x value formula

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90th Percentile of z scoresFigure 5.20

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Graph showing both the binomial probability histogram and the

normal distribution

Figure 5.13

The Normal Approximation of the Binomial

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The Normal Approximation of the Binomial

• Computing Probabilities for Any Normally Distributed Variable Method– Calculate mean and standard deviation– Apply continuity correction factor (±0.5)– Convert x values to z scores– Calculate area under standard normal curve

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End of Chapter 5

Probability Distributions