1. Introduction
Definition.
PROBABILITY is the study of CHANCE: a certain random process is given (such as
rolling a die or spinning a roulette wheel), and we want to know the chance of
various outcomes.
A. The CHANCE of a particular event is the
percentage of time that event is expected to occur if the same random process
is repeated over and over under the same circumstances. For example, the chance
that "a fair die comes up showing 1" is 1/6. (Note this definition implies the basic
process if replicable; if the process cannot be repeated, using the word
"chance" is not correct. )
B. Chances
can take values from 0% to 100%.
C. The chance of a particular event (say rolling a "1" using a fair die) is equal to 100% minus the chance of the event not happening (i.e. 100% minus not rolling a 1). This is a "complement". Where have you seen 100% before? (think Chapter 5)
3. Calculating Basic Probabilities:
the number of outcomes of interest ---------------------------------- = probability of an eventtotal possible number of outcomes
Using this formula requires that you know how many outcomes to expect. Example, Problem 7, Exercise Set C. A coin is tossed 3 times. What is the chance of getting at least 1 head? You need to know what can possibly happen before you can estimate the chance of getting at least 1 (that means 1, 2, or 3) heads.
4.
Sampling
(p. 225)
Here,
Freedman introduces you to the concept of sampling or "drawing" at
random from a "box". This
image will be important through the remainder of the text.
SAMPLING
WITH REPLACEMENT -- a situation where the total possible number of outcomes
remains the same with every random draw.
SAMPLING WITHOUT REPLACEMENT -- a situation where the total possible number of outcomes can change with every random draw.
5.
Multiplication,
Conditional Probabilities and Independence
The
multiplication rule: the chance that two (or more) things happen equals the
chance that the first happens multiplied by the conditional chance that the
second happens given that the first has happened. And you could extend this to the third, the fourth, etc.
Independence
is possible when Sampling with Replacement (13.4) and basically when 2 or more
events occur, one has no effect on the others occurring and vice versa. In
other words, (from page 230) if A happens, it does not change the chance of B
happening, etc.. To find out the chance that the two or more things happen, you
can multiply their unconditional probabilities.
Conditional
Probabilities and Sampling without Replacement (13.2). To help you, think of a box with 5 tickets
in it labeled 1,2,3,4,5. If I pull out a ticket, what is the chance that it is
a 4? (answer: 1/5)
If I DO NOT put the ticket back in (suppose it was a 4) and then pull another
out, what is the chance that it is a 3? (answer: 1/4 note that the chance of
has changed) selecting a 3 is not independent. It is dependent where selecting
a 4 first had some effect on selecting a 3 afterwards.
You
might think of this situation as selecting a ticket and then not replacing it
PUTS A CONDITION on the next ticket selected.
PRINCIPLE: In the situation where outcomes are independent,
you multiply unconditional probabilities. If the outcomes are dependent (i.e.
situations like sampling without replacement) then multiply conditional
probabilities.