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.
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?
- 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
- 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)
- 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)
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
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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