Introduction to Probability

The study of descriptive statistics was concerned with what has occurred, probability is concerned with what will occur. Many of the concepts are the same, although some of the vocabulary changes. Descriptive statistics is concerned with (relative) frequency in the past, probability with (relative) frequency in the future.

Vocabulary
Axioms
Where do probabilities come from?

Vocabulary

Experiment: something which generates an outcome (e.g., pick a card, roll a die, weigh a student, look outdoors)
Outcome (also called simple event): result of an experiment (e.g., jack of spades, 3 pips, 145 pounds, partly cloudy)
Sample space (denoted by S): set of all possible outcomes of an experiment (e.g., for picking a card there are 52 possible outcomes, hence 52 points in the sample space)
Event: a set of outcomes, or equivalently, a subset of the sample space (e.g., for picking a card, events include getting a spade, getting a deuce, getting a face card)

N.B.: Sometimes identifying outcomes is subtle. If you roll a pair of dice, is the total number of pips, the pair of values on the two dice, or the ordered pair of values on the two dice the outcome?

Axioms

A probability space entails that a probability be assigned to each outcome.

The probability of each outcome [denoted P(o_i), where o_i is the ith outcome] is between 0 and 1, inclusive.
The probability of an event is the sum of the probabilities of the outcomes (simple events) in the Event.
P(S)=1; Something has to happen, the probability of the sample space is 1.

Where do probabilities come from?

Probabilities may be given, often in the form of a table. For example, if an experiment has three possible outcomes: Apple, Banana, and Cherry, one might be given the following table:
```
   o_i |  A  |  B  |  C                                    
   ------------------------                                           
   p_i |  .5 |  .3 |  .2                       
 
```
N.B.: Even if the .2 entry had been missing, you would have been able to figure it out, since probabilities sum to 1.
Probabilities my be historical, if it has rained during 1/3 of the days in June during the past, one may say that the probability of rain for a day in June is 1/3.
Probabilities may be theoretical, if a die is fair (and there is any justice in the universe), since there are six possible outcomes, the probability of getting 3 pips on the top face is 1/6.

Competencies: Given the following (incomplete) table of probabilities associated with rolling an unfair die:

   o_i |  1  |  2  |   3  |  4  |  5  |  6  |                                   
   ------------------------------------------  
   p_i | .2  | .1  |  .3  | .1  |  ?  | .1  |

What is the probability of rolling a 5?
What is the probability of rolling an even number (2 or 4 or 6)?

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Questions?