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Lecture notes study book Understanding Probability of Henk Tijms - ISBN: 9781139511070 (Probability)
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Probability 145A. INTRODUCTION
Chance and Uncertainty
"Probability" is one of those ideas about which we all have some notion, but many of these
notions are not very definite. Initially, we will not spend our time trying to get an exact
definition, but will confine ourselves to the task of grasping the idea generally and seeing
how it can be used. Other words which convey much the same idea are "chance" and
"likelihood". Just as there is a scale of, say, temperature, because some things are hotter
than others, so there is a scale of probability, because some things are more probable than
others. Snow is more likely to fall in winter than in summer; a healthy person has more
chance of surviving an attack of influenza than an unhealthy person.
There is, note, some uncertainty in these matters. Most things in real life are uncertain to
some degree or other, and it is for this reason that the theory of probability is of great
practical value. It is the branch of mathematics which deals specifically with matters of
uncertainty, and with assigning a value between zero and one to measure how likely it is that
an event will occur. For the purpose of learning the theory, it is necessary to start with simple
things like coin tossing and dice throwing, which may seem a bit remote from business and
industrial life, but which will help you understand the more practical applications.
Choosing a Scale
First of all, let us see if we can introduce some precision into our vague ideas of probability.
Some things are very unlikely to happen. We may say that they are very improbable, or that
they have a very low probability of happening. Some things are so improbable that they are
absolutely impossible. Their probability is so low that it can be considered to be zero. This
immediately suggests to us that we can give a numerical value to at least one point on the
scale of probabilities. For impossible things, such as pigs flying unaided, the probability is
zero. By similarly considering things which are absolutely certain to occur, we can fix the top
end of the scale at 100%. For example, the probability that every living person will eventually
die is 100%.
It is often more convenient to talk in terms of proportions than percentages, and so we say
that for absolute certainty the probability is 1. In mathematical subjects we use symbols
rather than words, so we indicate probability by the letter p and we say that the probability
scale runs from p 0 to p 1 (so p can never be greater than 1).
Degrees of Probability
For things which may or may not happen, the probability of them happening obviously lies
somewhere between 0 and 1.
First, consider tossing a coin. When it falls, it will show either heads or tails. As the coin is a
fairly symmetrical object and as we know no reason why it should fall one way rather than
the other, then we feel intuitively that there is an equal chance (or, as we sometimes say, a
50/50 chance) that it will fall either way. For a situation of equal chance, the probability must
lie exactly halfway along the scale, and so the probability that a coin will fall heads is 1/2 (or
p 0.5). For tails, p is also 0.5.
Next, consider rolling a six-sided die as used in gambling games. Here again this is a fairly
symmetrical object, and we know of no special reason why one side should fall uppermost
more than any other. In this case there is a 1 in 6 chance that any specified face will fall
uppermost, since there are 6 faces on a cube. So the probability for any one face is 1/6 (p
0.167).
As a third and final example, imagine a box containing 100 beads of which 23 are black and
77 are white. If we pick one bead out of the box at random (blindfold and with the box well
,146 Probability
shaken up) what is the probability that we will draw a black bead? We have 23 chances out
of 100, so the probability is 23/100 (or p 0.23).
Probabilities of this kind, where we can assess them from prior knowledge of the situation,
are called a priori probabilities.
In many cases in real life it is not possible to assess a priori probabilities, and so we must
look for some other method. What is the probability that a certain drug will cure a person of a
specific disease? What is the probability that a bus will complete its journey without having
picked up a specified number of passengers? These are the types of probabilities that cannot
be assessed a priori. In such cases we have to resort to experiment. We count the relative
frequency with which an event occurs, and we call that the probability. In the drug example,
we count the number of cured patients as a proportion of the total number of treated patients.
The probability of cure is then taken to be:
number of patients cured
p
number of patients treated
In a case like this, the value we get is only an estimate. If we have more patients, we get a
better estimate. This means that there is the problem of how many events to count before the
probability can be estimated accurately. The problem is the same as that faced in sample
surveys. Probabilities assessed in this way, as observed proportions or relative frequencies,
are called empirical probabilities.
In cases where it is not possible to assign a priori probabilities and there is no empirical data
available to enable empirical probabilities to be computed, probabilities may simply have to
be based on people's experiences and personal opinions. Such probabilities are called
subjective probabilities, and in such cases, it is normally advisable to consult an expert in the
field. For example, if a business manager wishes to know the probability that interest rates
will rise over the next three months, he or she would be advised to consult a financial
economist to gain an expert opinion.
B. TWO LAWS OF PROBABILITY
Addition Law for Mutually Exclusive Events
If a coin is tossed, the probability that it will fall heads is 0.5. The probability that it will fall tails
is also 0.5. It is certain to fall on one side or the other, so the probability that it will fall either
heads or tails is 1. This is, of course, the sum of the two separate probabilities of 0.5. This is
an example of the addition law of probability. We state the addition law as:
"The probability that one or other of several mutually exclusive events
will occur is the sum of the probabilities of the several separate
events."
Note the expression "mutually exclusive". This law of probability applies only in cases where
the occurrence of one event excludes the possibility of any of the others. We shall see later
how to modify the addition law when events are not mutually exclusive.
Heads automatically excludes the possibility of tails. On the throw of a die, a six excludes all
other possibilities. In fact, all the sides of a die are mutually exclusive; the occurrence of any
one of them as the top face necessarily excludes all the others.
, Probability 147
Example:
What is the probability that when a die is thrown, the uppermost face will be either a two or a
three?
The probability that it will be two is 1/6.
The probability that it will be three is 1/6.
Because the two, the three, and all the other faces are mutually exclusive, we can use the
addition law to get the answer, which is 2/6, i.e. 1/6 1/6, or 1/3.
You may find it helpful to remember that we use the addition law when we are asking for a
probability in an either/or situation.
Complementary Events
An event either occurs or does not occur, i.e. we are certain that one or other of these two
situations holds. Thus the probability of an event occurring plus the probability of the event
not occurring must add up to one, that is:
P(A) P(not A) 1 (a)
where P(A) stands for the probability of event A occurring.
A1 or A is often used as a symbol for "not A". "A" and "not A" are referred to as
complementary events. The relationship (a) is very useful, as it is often easier to find the
probability of an event not occurring than to find the probability that it does occur. Using (a)
we can always find P(A) by subtracting P(not A) from one.
Example:
What is the probability of a score greater than one when one die is thrown once?
Method 1:
Probability of a score greater than 1 P(score 1)
P(score not 1)
1 P(score 1), using (a)
1 5
1
6 6
Method 2:
Probability of a score greater than 1 P(score 1)
P(2 or 3 or 4 or 5 or 6)
P(2) P(3) P( 4) P(5) P(6) using addition law
1 1 1 1 1 5
as before.
6 6 6 6 6 6
Multiplication Law for Independent Events
Now suppose that two coins are tossed. The probability that the first coin will show heads is
0.5, and the probability that the second coin will show heads is also 0.5. But what is the
probability that both coins will show heads? We cannot use the addition law because the two
events are not mutually exclusive – a first coin landing heads does not prevent a second coin
landing heads. If you did try to apply the addition law, you would get 0.5 0.5, which is 1.0,
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