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Basic Probability Theory
for Biomedical Engineers

i

Page 2

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in

any form or by any means—electronic, mechanical, photocopy, recording, or any other except for brief quotations

in printed reviews, without the prior permission of the publisher.

Basic Probability Theory for Biomedical Engineers

John D. Enderle, David C. Farden, Daniel J. Krause

www.morganclaypool.com

ISBN: 1598290606 paper

ISBN: 9781598290608 paper

ISBN: 1598290614 ebook

ISBN: 9781598290615 ebook

DOI10.2200/S00037ED1V01Y200606BME005

A Publication in the Morgan & Claypool Publishers’ series

SYNTHESIS LECTURES ON BIOMEDICAL ENGINEERING

Lecture #5

Series Editor and Affliation: John D. Enderle, University of Connecticut

1930-0328 Print

1930-0336 Electronic

First Edition

10 9 8 7 6 5 4 3 2 1

Printed in the United States of America

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58 BASIC PROBABILITY THEORY FOR BIOMEDICAL ENGINEERS

Drill Problem 1.8.4. Four boxes contain the following quantity of marbles.

RED BLUE GREEN

Box 1 6 3 2

Box 2 5 4 0

Box 3 3 3 4

Box 4 2 9 7

A box is selected at random and the marble selected is green. Determine the probability that: (a) box 1

was selected, (b) box 2 was selected, (c) box 3 was selected, (d) box 4 was selected.

1.9 SUMMARY
In this chapter, we have studied the fundamentals of probability theory upon which all of

our future work is based. Our discussion began with the preliminary topics of set theory, the

sample space for an experiment, and combinatorial mathematics. Using set theory notation,

we developed a theory of probability which is summarized by the probability space (S,F, P ),
where S is the experimental outcome space, F is a �-field of subsets of S (F is the event space),
and P is a probability measure which assigns a probability to each event in the event space. It is

important to emphasize that the axioms of probability do not dictate the choice of probability

measure P . Rather, they provide conditions that the probability measure must satisfy. For the

countable outcome spaces that we have seen so far, the probabilities are assigned using either

the classical or the relative frequency method.

Notation for joint probabilities has been defined. The concept of joint probability is

useful for studying combined experiments. Joint probabilities may always be defined in terms

of intersections of events.

We defined two events A and B to be independent iff (if and only if )

P (A ∩ B) = P (A)P (B).

The extension to multiple events was found to be straightforward.

Next, we introduced the definition for conditional probability as the probability of event

B, given event A occurred

P (B|A) = P (A ∩ B)
P (A)

,

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INTRODUCTION 59

(a)

H

T

T

H

H

T HT

TH

TT

HH 2

1

1

0

Outcome k

0.5

0.5

0.5

0.5

0.5

0.5

0

1

2

k

0.25
0.5

0.25

(b)

FIGURE 1.16: Partial tree diagram for Example 1.9.1

provided that P (A) = 0. The extension of the definition of conditional probability to multiple
events involved no new concepts, just application of the axioms of probability. We next presented

the Theorem of Total Probability and Bayes’ Theorem.

Thus, this chapter presented the basic concepts of probability theory and illustrated

techniques for solving problems involving a countable outcome space. The solution, as we have

seen, typically involves the following steps:

1. List or otherwise describe events of interest in an event space F,
2. Assignment/computation of probabilities, and

3. Solve for the desired event probability.

The following example illustrates each of these steps in the solution.

Example 1.9.1. An experiment begins by rolling a fair tetrahedral die with faces labeled 0, 1, 2, and

3. The outcome of this roll determines the number of times a fair coin is to be flipped.

(a) Set up a probability tree for the event space associating the outcome of the die toss and the

(b) If there were two heads tossed, then what is the probability of a 2 resulting from the die toss?

Solution. Let n be the value of the die throw, and k be the total number of heads resulting

from the coin flips.

(a) Since it is fairly difficult to draw the probability tree for this experiment directly, we shall

develop it in stages. We first draw a partial probability tree shown in Fig. 1.16(a) for

the case in which the die outcome is two and the coin is flipped twice. This probability

tree can be compressed into a more efficient event space representation as shown in

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125

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