ACMS10145 Lecture Notes - Lecture 4: Prope, Sample Space, Blu-Ray
19 Feb 2019
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Chapter 4: Introduction to Probability
In Chapter 4 we learn to develop our ability to think probabilistically. We will focus on a
number of definitions, then on “rules” of probability. I believe it helps to think of these
things as tools in a toolbox that we can use to attack problems, rather than fixed steps, or
a “cookbook” that we must follow.
The word “probability” does not have a single direct definition for practical application.
Consider the following questions:
• What are the chances that sales will decrease if we increase prices?
• What is the probability that buying 1,000 shares of the Ford Motor Company will
yield a profit in the next year?
• How likely is it that project will be finished on time?
• What is the likelihood a new assembly method will increase productivity?
How do these questions differ from each other?
We’ll try to see that probability can illuminate our understanding of the world. Some
concepts will be easily grasped by the intuition, and some will not.
Probability is a numerical measure of the likelihood that an event will occur.
Sections 4.1-4.2: Counting Rules, Assigning Probabilities and their Properties
Almost everyone has at least a basic intuitive understanding that probabilities take values
between 0 and 1 (or, 0 and 100% if we want to use percent instead of proportions. Either is
fine for this class.) An event with a probability near 0 is said to be very unlikely to occur,
while an event with a probability near 1 is said to be very likely to occur. We’ll formalize
these ideas mathematically.
Some basic definitions:
• An experiment is an act or process of watching that leads to a single outcome that
cannot be predicted with certainty. An experiment can be complex, such as a
chemistry experiment, or it may be simple, such as rolling a die or flipping a coin.
• A sample point is the most basic outcome of an experiment, and the sample space is
the collection of all its sample points.
• An event is a specific collection of sample points.
• A sample space is a collection of all possible outcomes of an experiment.
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These concepts are best illustrated via simple examples.
Eg1: Consider an experiment in which we are simply tossing a single die one time. We can
roll a “1”, “2”, “3”, “4”, “5”, or “6”. Each of these is a sample point, and the collection of all
of them is a sample space, which is denoted with brackets: {1, 2, 3, 4, 5, 6}. “Rolling an even
number” is an example of an event, i.e., it corresponds to the subset of the sample space
given by {2, 4, 6}.
1. Multiplication principle: Suppose an experiment E1 has n1 outcomes and for each of
these outcomes an experiment E2 has n2 possible outcomes. The composite
experiment E1E2 that consists of performing first E1 and then E2 has n1n2 possible
outcomes.
Eg2: If you can choose any of these 3 ice cream flavors: chocolate, strawberry,
vanilla, and any of these 4 toppings: caramel, M&M’s, nuts, and strawberries, how
many options can you choose?
Eg3: Suppose we have n people to fill in for n jobs. How many arrangements are
possible?
2. Definition: Each of the n! arrangements of n different objects is called a
permutation of the n objects.
However, if only r positions are to be filled by n different people, then the number
of possible ordered arrangements is:
nPr= n(n-1)…(n-r+1)=
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Eg4: Suppose we have 15 people apply for 3 jobs, and suppose they are all equally
qualified. How many arrangements are possible?
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Document Summary
In chapter 4 we learn to develop our ability to think probabilistically. We will focus on a number of definitions, then on rules of probability. I believe it helps to think of these things as tools in a toolbox that we can use to attack problems, rather than fixed steps, or a cookbook that we must follow. The word probability does not have a single direct definition for practical application. We"ll try to see that probability can illuminate our understanding of the world. Some concepts will be easily grasped by the intuition, and some will not. Probability is a numerical measure of the likelihood that an event will occur. Sections 4. 1-4. 2: counting rules, assigning probabilities and their properties. Almost everyone has at least a basic intuitive understanding that probabilities take values between 0 and 1 (or, 0 and 100% if we want to use percent instead of proportions.
