© 2008 McGraw-Hill Higher Education The Statistical Imagination Chapter 6. Probability Theory and the Normal Probability Distribution

© 2008 McGraw-Hill Higher Education

The Statistical Imagination

• Chapter 6. Probability Theory and the Normal Probability Distribution


Probability Theory

• Probability theory is the analysis and understanding of chance occurrences


What is a Probability?

• A probability is a specification of how frequently a particular event of interest is likely to occur over a large number of trials

• Probability of success is the probability of an event occurring

• Probability of failure is the probability of an event not occurring


The Basic Formula for Calculating a Probability

• p [of success] = the number of successes divided by the number of trials or possible outcomes, where p [of success] = the probability of “the event of interest"


Basic Rules of Probability Theory

• There are five basic rules of probability that underlie all calculations of probabilities


Probability Rule 1: Probabilities Always Range Between 0 and 1

• Since probabilities are proportions of a total number of possible events, the lower limit is a proportion of zero (or a percentage of 0%)

• A probability of zero means the event cannot happen, e.g., p [of an individual making a free-standing leap of 30 feet into the air] = 0

• A probability of 1.00 (or 100%) means the event will absolutely happen, e.g., p [that a raw egg will break if struck with a hammer ] = 1.00


Probability Rule 2: The Addition Rule for Alternative Events

• An alternative event is where there is more than one outcome that makes for success

• The addition rule states that the probability of alternative events is equal to the sum of the probabilities of the individual events

• For example, for a deck of 52 playing cards: p [ace or jack] = p [ace] + p [jack]

• The word or is a cue to add probabilities; substitute a plus sign for the word or


Probability Rule 3: Adjust for Joint Occurrences

• Sometimes a single outcome is successful in more than one way

• An example: What is the probability that a randomly selected student in the class is male or single? A single-male fits both criteria

• We call “single-male” a joint occurrence an event that double counts success

• When calculating the probability of alternative events, search for joint occurrences and subtract the double counts


Probability Rule 4: The Multiplication Rule

• A compound event is a multiple-part event, such as flipping a coin twice

• The multiplication rule states that the probability of a compound event is equal to the multiple of the probabilities of the separate parts of the event

• E.g., p [queen then jack] = p [queen] • p [jack]• By multiplying, we extract the number of

successes in the numerator and the number of possible outcomes in the denominator


Probability Rule 5: Replacement and Compound Events

• With compound events we must stipulate whether replacement is to take place. For example, in drawing a queen and then a jack from a deck of cards, are we to replace the queen before drawing for the jack?

• The probability “with replacement” will compute differently than “without replacement”


Using the Normal Curve as a Probability Distribution

• With an interval/ratio variable that is normally distributed, we can compute Z-scores and use them to determine the proportion of a population’s scores falling between any two scores in the distribution

• Partitioning the normal curve refers to computing Z-scores and using them to determine any area under the curve


Three Ways to Interpret the Symbol, p

1. A distributional interpretation that describes the result in relation to the distribution of scores in a population or sample

2. A graphical interpretation that describes the proportion of the area under a normal curve

3. A probabilistic interpretation that describes the probability of a single random drawing of a subject from this population


Procedure for Partitioning Areas Under the Normal Curve

1. Draw and label the normal curve stipulating values of X and corresponding values of Z

2. Identify and shade the target area ( p ) under the curve

3. Compute Z-scores4. Locate a Z-score in column A of the

normal curve table5. Obtain the probability ( p ) from either

column B or column C


Information Provided in the Normal Curve Table

• Column A contains Z-scores for one side of the curve or the other

• Column B provides areas under the curve ( p ) from the mean of X to the Z-score in column A

• Column C provides areas under the curve from the Z-score in column A out into the tail


Important Considerations in Partitioning Normal Curves

• The variable must be of interval/ratio level of measurement

• The sample and population must be normally distributed

• Always draw the curve and its target area to avoid mistakes in reading the normal curve table


Percentiles and the Normal Curve

• A percentile rank is the percentage of a sample or population that falls at or below a specified value of a variable

• If a distribution of scores is normal in shape, then the normal curve and Z-scores can be used to quickly calculate percentile ranks


Statistical Follies

• The Gambler’s Fallacy is the notion that past gaming events, such as the roll of dice in the casino game, Craps, are affected (or dependent upon) past events

• E.g., It is fallacious to think that because a coin came up heads three times in a row that tails is bound to come up on the next toss

• The tosses are independent of one another

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© 2008 McGraw-Hill Higher Education The Statistical Imagination Chapter 6. Probability Theory and the Normal Probability Distribution