5.1 Probability of Simple Events

5.1 Probability of Simple Events
1. Probability theory provides a mathematical basis for inferential statistics.
2. In statistics, an experiment is a process that can be repeated and whose results are
uncertain. (Examples: flip a coin; draw a card from a deck; draw six cards from a
deck)
3. An outcome is one single possible result of an experiment. (Example: obtaining
a head when a coin is tossed.) If an experiment is performed once, an outcome
will result. We call a single outcome a simple event and denote it as e i
.
4. The sample space of a probability experiment is the set of all possible outcomes
that may occur when the experiment is performed. It is the list of all possible
outcomes of an experiment. (Example: If the experiment is to flip a coin, then
the sample space is {H, T}.)
5. An event, denoted as E, is any collection of outcomes. It may consist of one or
more simple events. (An event could also be empty.)
It is a subset of the sample space.
An event may or may not occur when the experiment is performed.
If an experiment is performed and the result is one of the outcomes in the
event, we say that the event has occurred.
6. The probability of an outcome measures the likelihood of that outcome
occurring. It is also the proportion of times the outcome will occur if the
experiment that generates it is performed a large number of times. It can be
expressed as a decimal, a fraction, or a percent. (The probability of obtaining a
head when a coin is flipped is 1/2.)
7. Law of Large Numbers: (See page 224.) As the number of repetitions of a
probability experiment increases, the proportion with which a certain outcome is
observed gets closer to the probability of the outcome.
8. Denote the probability of an event E as P(E). Denote the sample space as S.
9. Basic Properties of Probabilities: (See page 225.)
Property 1: The probability of an event E, is always between 0 and 1,
inclusive. That is 0 ≤ P(E ) ≤ 1.

Property 2: The sum of all of the probabilities of all simple events is 1.
That is, if S = {e 1
, e 2
, ...,e n
} is the sample space of an experiment, then
P(e 1
) + P(e 2
) + ...+ P(e n
) =1.
10. A probability model lists each possible outcome together with its probability.
The probability model must satisfy properties 1 and 2 above.
11. The probability of an event that cannot occur is 0. (An event that cannot occur is
called an impossible event.)
The probability of an event that must occur is 1. (An event that must occur is
called a certain event.)
12. If an event has a probability near 0 (or 0%), it is not very likely to occur. If an
event has a probability near 1 (or 100%), it is very likely to occur.
13. An unusual event is an event that has a low probability of occurring. (Typically,
but not always, it has a probability less than 5%.)
14. There are three methods for determining probabilities, the classical method, the
empirical method and the subjective method.
15. Empirical Probabilities: (Based on counting the number of times an outcome
occurs when as experiment is performed repeatedly.) See page 226.
The probability of an event E is approximately the number of times event E
occurs divided by the number of repetitions of the experiment.
frequency of E
P(E ) ≈ relative frequency of E =
number of trials of the experiment
16. Note that probabilities calculated using the empirical approach are approximate.
By the Law of Large Numbers, probabilities calculated with this approach will
become closer to the true probability as the number of trials increases.
17. Classical Method: Probability for Equally Likely Outcomes: (See page 227.)
An experiment is said to have equally likely outcomes if each outcome has an
equal chance of occurring.
Let n be the number of possible equally likely outcomes of an experiment or
game. Let m be the number of ways that a specified event can occur. Then the
probability that the event E, occurs is m/n.
P(E) =
Number of ways the event can occur
Total number of possible outcomes = m n .

P(E) = N(E) , where N(E) is the number of outcomes in the event E,
N(S)
and N(S) is the number of outcomes in the sample space.
18. Subjective probabilities are probabilities obtained based upon an educated guess.
Each person could have a different probability for an event, based on the way they
interpret the relevant information. Therefore, subjective probabilities should be
interpreted with skepticism.