Statistics 215 Lab Materials
Probability
We will introduce some of the concepts associated with probability. What we have covered so far has been
descriptive statistics, that is, summaries of the data. The rest of this book is devoted to inferential statistics
or decision making from data. This chapter begins our steps in that direction. In general probability is the
study of the chance that an event or events will happen, assuming we don’t yet know the outcome. Another
way to think of probability is the chance that one or more of the possible outcomes from all the possible
outcomes is the outcome. In this way, we can think of probability as the chance a particular sample is the
resulting subset from a population. The key to all of this is that each event has a different outcome.
Consequently there is variation.
Preliminary Definitions
We begin this section with some definitions.
Definition: A trial is the act or process that leads to a single outcome that cannot be predicted with perfect
certainty.
Definition: An experiment is one or more trials
Definition: An outcome of an experiment or trial is the single result that cannot be broken into smaller
events.
Definition: An event is a collection of one or more outcomes
Definition: A sample space is the collection of all possible outcomes, often denoted by S.
Probability is often thought to deal with cards, dice and other games. And it is true that the original
mathematical studies of probability began as studies of gambling and games of chance. However,
probability can and is used in a myriad of ways outside of this arena. Probability is used a great deal by
economists, as well as by military strategists. More informally, it is used by most of us in our everyday
lives. Is it faster to take route A home or route B? We make a decision based on probabilities we have in
our mind from taking both routes. Of course, it is possible that today route A may be faster, while
tomorrow route B will be faster, just like the first time I flip a coin I might get a heads and the second I
might get a tails.
Example: Rolling a six-sided.
Suppose that we roll a die and record the number on top. The sample space S = {1,2,3,4,5,6}. This is a
single trial with only one outcome. We might be interested in the event that the roll produces a number
more than 3. That event can be broken into the simple events of getting a 4, getting a 5 or getting a 6.
Example: Sales of Cap’n Crunch at Sam’s Market
Suppose that we record the number of boxes of Cap’n Crunch that are sold at Sam’s Market each day for
the next 7 days. This is an experiment with 7 different trials. For each trial the sample space is S= {0, 1, 2,
3, 4, …}. One event of interest might be that Sam’s sells less than 4 boxes on Tuesday. This event is
comprised of simpler events: selling 0 boxes on Tuesday, selling 1 box on Tuesday , selling 2 boxes on
Tuesday, selling 3 boxes on Tuesday.
Definition of probability
There are many ways to define the probability of an event. We will discuss two of them. Both of these
definitions are ways to define probability for simple events.
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Statistics 215 Lab Materials
Definition: the relative frequency definition of probability says that the probability of a simple event is the
number of times that the event occurred divided by the number of trials for a very long series of repetitions
Definition: the subjective frequency definition of probability says that the probability of a simple event is
what you think the chance of the event will be using any prior knowledge of the experiment.
There are drawbacks to each of these definitions. The relative frequency definition requires that a long
series of repetitions be performed. For events like the 1999 West Virginia versus Virginia Tech football
game or the 2000 presidential election or the finals of the 2000 Olympic women’s heptathlon we don’t have
a series of repetitions and we never will. Consequently it is hard to have a relative frequency definition of
those events. Likewise there is the question of how many repetitions makes a very long series. The
subjective frequency definition has the difficulty that two different people could choose two different
probabilities for the same event. If I say the probability of it raining tomorrow is 75% and you say that is is
56%, we can’t both be right. The good news is that once probabilities for simple events have been
determined, the rest of the rules for probabilities are the same. Throughout the rest of this book, we will
use the relative frequency definition.
Probability of simple events
There are three basic rules that all definitions of probability must follow in order to be valid.
They are:
1. All simple events must have probabilities between 0 and 1.
2. The sum of the probabilities of all simple events in the sample space must be 1.
3. The probability for an event is the sum of the probabilities of the simple events that make it up.
A couple of comments are necessary. First a probability of 0, or 0%, means the event cannot happen. The
probability of rolling a typical six-sided die and getting 4.32 is 0. Zero is also used to describe the
probability of events that did not happen. The probability of Walter Mondale winning the 1984 presidential
election is 0. Second, a probability of 1, or 100%, means the outcome will always happen. 1 is also used to
described events that occurred in the past. With probability 1, George Washington was the first president
of the United States of America.
Some notation:
Events like variable will be denoted with a capital letter.
Example:
A=sales of Cap’n Crunch this week at Sam’s Market will be more than 20
H=over 50 people will attend the next Statistics 215 lecture
Y= you toss a coin and get heads
P(F) will represent the probability that event F occurs. So using the above example P(H) is the probability
that over 50 people will attend the next Statistics 215 lecture.
Calculating probabilities
There are two possible ways to calculate events that we will worry about in this book.
1.
Theoretical Probability. (For equally likely outcomes.) If each trial can result in one of r equally
likely simple outcomes and m of those could cause event X to happen, then the theoretical
probability of event X is
P( X )=
2.
r
.
k
Experimental Probability. Suppose that an experiment consist of n trials, and k of these trials
results in event X. then the theoretical probability of event X is
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Statistics 215 Lab Materials
k
Pˆ (X)= .
n
Note the difference (and similarity) in the notation for theoretical probability and experimental probability.
The experimental probability
of an event is also called the empirical probability of an event.
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The experimental probability of an event can be used to estimate the theoretical probability of an event.
Example:
Suppose that we have a bag of chips that are different color. When we draw a chip from the bag, assume
that we cannot tell the difference in the color of the chips by feeling the chips, so that the chips are drawn at
random.
This particular bag has
15 red chips
10 blue chips
25 white chips
If I selected a chip from the bag at random, what is the chance that I get a blue chip?
P(blue chip) = 10/50= 0.20, since r=10 chips and k = 50 total chips.
If I selected a chip from the bag at random, what is the chance that I get a red chip?
P(red chip) = 15/50 = 0.30, since there are 15 red chips and 50 chips total. So r = 15 and k = 50.
If I selected a chip from the bag at random, what is the chance that I get a white chip?
P(white chip) = 25/50 = 0.50, since r = 25 chips and k = 50 chips.
Example:
Suppose we have a different bag with chips. Each chip is the same size but each one has a different color
and a different symbol on them.
This bag has 5 red triangle chips, 10 red square chips, 5 red circle chips, and
10 white triangle chips, 10 white square chips, 10 white circle chips and
5 blue triangle chips, 5 blue square chips.
If I selected a chip from the bag at random, what is the chance that I get a red chip?
P(red chip) = 20/60 = 0.333, since r = 20 chips and k = 60 chips.
If I selected a chip from the bag at random, what is the chance that I get a chip with a square on it?
P(square chip) = 25/60 = -.417, since there are 25 chips with squares on them and 60 chips total.
If I selected a chip from the bag at random, what is the chance that I get a chip with a circle on it?
P(circle chip) = 15/60 = 0.250, since there are 15 chips with circles on them and 60 chips total, then r = 15
and k = 60.
Example:
I toss a fair coin 100 times and observe the number of heads and the number of tails that result. If I observe
48 heads then
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Statistics 215 Lab Materials
48
Pˆ (Head) =
= 0.48
100
and
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52
Pˆ (Tail) =
= 0.52
100
€
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