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Chapter 6: Probability

Section 6.2:   Probability Models

A Little Probability Background
The mathematics of chance/randomness is called Probability.

Chance behavior is unpredictable in the short run, but has a regular and predictable pattern in the long run.   For example, toss a coin –in the long run, about ½ will be heads P(heads) = ½.  

Samples are random – we are uncertain who will be chosen, but overall everyone has an equal chance of being selected.

Definition of Randomness and Probability. We call a phenomenon random if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions. The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions

Examples. If we roll a 6 sided die, the probability of getting a 4 is 1/6. Flip a (fair) coin and the probability of “heads” is 1/2.

Note:   Probabilities can be expressed as a fraction, decimal, or as a percent.   The most common ways are as fractions or decimals.


Myths of Probabilities  

Myth of short-run regularity – Remember that the patterns of probability apply to the long run.

Myth of the law of averages --   Often random events are independent of previous events.   This means that, knowing the outcome of one trial does not changed the probabilities for the outcomes of any other trials.   For instance just because a fair coin has been flipped heads six times in a row, it is NOT more likely that the seventh flip will be tails.


Studies have shown, as well, that this applies to athletics. For instance, just because one player has made five free throws in the beginning of a game, doesn’t mean that he has an increased probability of making(or missing) his sixth free-throw.

Probability Models
The sample space S of a random phenomenon is the list of all possible outcomes.

An event is any collection of outcomes in the sample...