Download Introduction to Probability

Introduction to Probability
Introduction to Probability
RANDOM EXPERIMENT - a process leading to observation of a phenomenon such that
(a). there is more than one possible outcome to the experiment, and
(b). the outcome that actually occurs cannot be predicted with certainty.
e.g. One card is selected at random from the 52 cards in a standard deck. There are 52
possible outcomes, and one cannot predict with certainty which one (card) will occur.
OUTCOME or ELEMENT - one of the possible outcomes when a random experiment is
performed. (i.e. e )
e.g. Each of the 52 cards is an element or outcome.
SAMPLE SPACE - the set of all possible outcomes of a random experiment. (S = {e1, e2,…eN })
e.g. S = {the set of 52 cards in a standard deck}
TRIAL - one repetition of an experiment.
e.g. Drawing one card. Drawing another card would be a second trial.
EVENT - a subset of the sample space. ( E, A)
e.g. D = {card drawn is a Diamond}
SIMPLE EVENT - an event consisting of one single outcome or element of the sample space.
(i.e. of the form {e})
e.g. A = {card drawn is the Ace of Spades}
CERTAIN or SURE Event – an event that will definitely occur when the random experiment is
performed. It always amounts to the whole sample space S.
e.g. A card is drawn at random from a standard deck. Let C be the event “the card drawn
is red or black”. Since all cards in the deck are either red or black, this event must occur.
Also, C = S, since the elements of C are exactly the elements of S (i.e. the 52 cards).
IMPOSSIBLE or NULL Event – is an event that cannot occur. It always amounts to the null or
empty set, 
e.g. When a card is drawn at random from a standard deck, consider the event G being the
event “the card drawn is green”. Since none of the cards in the deck are green, this event is
impossible, and G =  = { }.
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Introduction to Probability
What is probability?
The probability of an event E is the likelihood or chance of the event happening or
occurring when a random experiment is performed, and it is represented as a number
between 0 and 1 inclusively.
MODELS FOR PROBABILITY
Probability is a function that operates on the events of a sample space.
If E is one event, its probability will be denoted by one of the following:
P(E) or P[E] or P{E}.
1. Classical Probability – Equally Likely Outcomes
Suppose the sample space S of a random experiment consists of N equally likely outcomes.
This might be written as #(S) = N.
For any simple event {e}, the probability of any simple event {e} (that consists of
one single outcome) is given as
1
Pe  .
N
If E is an event containing n equally likely outcomes - hat is, #(E) = n – then the
probability of event E is
PE  
# E  n
 .
# S  N
e.g. In drawing one card at random from a standard deck, the sample space S contains 52
1
12
 0.01923 and. PFace card  
 0.2308 .
equally likely cards. Hence PAce of Spades 
52
52
2. Relative Frequency Approach
If a random experiment is repeated M times (i.e. there are M repeated trials of the
experiment), and if the event E occurs on m of these trials, then the relative frequency with which
event E occurs is
m
R.F . 
.
M
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Introduction to Probability
The probability of E occurring may then be defined as the “long run relative frequency” with
m
which the event E occurs; that is, as the limit of R.F . 
as the number of trials M increases
M
without bound. This idea is written in mathematical form as
PE  
limit
M 
R.F . 
lim it  m 
lim  m 
 
 
M   M  M   M 
For example, what is the probability that “Heads” is observed when a fair coin is tossed
once? Using the Classical approach with N = 2 equally likely possibilities in the sample space,
1
the answer of PHeads   . What does this really mean? It obviously doesn’t mean that the
2
number of Heads observed is one-half. But it can be interpreted as meaning that Heads will be
observed one-half of the time (the relative frequency) when a coin is tossed repeatedly.
Consider an old fashioned thumbtack that has a circular head with a pointed metal piece
protruding from the centre of the head. When such an object is dropped on a table, it will
ultimately come to rest with the “point up” or with the “point down”. So the sample space of
possible outcomes is S = {point up, point down} consisting of two outcomes. Are they equally
likely? The answer may depend upon the sizes of the head and the pointed piece, and certainly
could be “no”. So how can the probability of a thumbtack falling “point up” be determined?
Using the Relative Frequency approach, one might drop such a thumbtack onto a table over and
over again and then, after a large number of trials, determine the relative frequency with which it
landed “point up”. If the number of trials is very large, interpreting this relative frequency as at
least an approximation to the required probability is reasonable. The actual probability is defined
as the limit of such values as the number of trials grows without bound. Of course, the true value
can never be obtained because the trials must continue forever!!!
3. Empirical Probability
Empirical probability is based on the idea of relative frequency discussed above. In this
case, probability is the relative frequency with which an event has occurred in a finite number of
trials (or observations).
For example, suppose that over the course of one summer the lengths (in centimeters) of a
certain type of fish (e.g. walleye or pickerel) are recorded at a small lake in northern
Saskatchewan. Suppose the results were as follows.
Length
Number
<20 cm
258
≥20 to <30
321
≥30 to <40
443
≥40 to <50
291
≥ 50 cm
101
Total
1414
What is the probability that a fish caught at this lake this past summer was at least 40 cm in
length? A total of 1414 fish were caught at this lake, and 291 + 101 = 392 of them were at least
40 cm in length. Hence the required probability is estimated as
392
PLength at least 40 cm 
 0.2772 .
1414
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Introduction to Probability
A second look at this procedure suggests that it is essentially equivalent to the Classical
probability idea. Here it is assumed that the 1414 fish caught over the summer are equally likely
outcomes, and that the event L = {length of fish caught is at least 40 cm}consists of 392 of these
equally likely outcomes.
4. Subjective Probability
There are situations in which a probability is wanted but the above ideas can’t be applied.
For example, what is the probability that it will snow in Saskatoon tomorrow? Evening weather
reports attempt to provide us with such probabilities. How are they arrived at?
The answer is that past experience (similar weather conditions having been observed
before), computer models to provide scientific analysis, and finally personal judgment in which an
“expert” assesses all of the available information to produce a number, are all used to come up
with an answer. Different experts might reach somewhat different answers after assessing the
information, and so a reported value is subjective.
This type of probability will not be considered as we might all come up with different
answers to a question!!
5. Odds
Sometimes the chance of an event occurring (or not) is expressed somewhat differently,
namely as odds for or against the event occurring. This method is essentially another way of
indicating Classical or equally likely probabilities.
Suppose that the sample space S of a random experiment is made up of N equally likely
outcomes and that, of these n correspond to an event E while the remaining m = N – n do not. The
probability of the event E occurring is then
# E  n
n
(1)
PE  


# S  N n  m
and the probability that the event E does not occur is
# E  m
m
.
(2)
PE  


# S  N n  m
The odds in favour of event E are then indicated as
n:m
which is read as “n to m” or “n is to m”. This indicates that, of a total of n + m equally likely
possibilities or chances, n of them correspond to event E happening and m correspond to event E
happening (i.e. to event E not happening).
The odds against an event E are indicated in the reverse order as
m:n
which is read as “m to n” or “m is to n”. This indicates that, of a total of n + m equally likely
possibilities or chances, m of them correspond to event E not happening (i.e. E happening) and n
correspond to event E happening.
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Note that, if the odds for or against an event are given, then the probability of that event
happening or not can be determined using the equations (1) or (2) given above. Conversely, if the
probabilities of the event happening or not are known, the odds are determined as follows.
Odds in favour of E happening:
P[E] : P[ E ]
Odds against E happening: P[ E ] : P[E] .
In either of these cases, the odds expressions are modified to involve the smallest possible positive
integers that express the same ratio.
Example:
Before writing a test in STATS 103.3, a student estimates her odds of obtaining a mark of
at least 80% in the course as 3:2 . Thus she is suggesting that, of 3 + 2 = 5 equally likely
possibilities, 3 of them correspond to her achieving such a mark and 2 of them do not. Therefore
she estimates that
3
2
PMark of at least 80%   0.60 and PMark less than 80%   0.40 .
5
5
After writing her first test the student reassesses her situation. She now thinks that the
probability of achieving a mark of at least 80% is 0.625 and hence the probability that she will not
do so is 0.375. Under these conditions, she now feels that the odds in favour of a mark of at least
80% can be stated as 0.625 : 0.375 . These should be changed to integers, and can be done so as
follows:
0.625 : 0.375 => 625 : 375 => 5 : 3 .
So her new odds in favour of a mark of at least 80% are 5 : 3 and her odds against a mark this
high are then 3 : 5.
BASIC PROPERTIES or AXIOMS OF PROBABILITY
For any random experiment, the following basic properties or axioms apply.
1. For any event E, 0  PE   1 . (Probability is a number between 0 and 1).
2. For the sample space S, PS   1 . (The event S is a sure event)
3. For any two distinct elements e and f of S, Pe or f   Pe  P f  .
(for distinct outcomes, probabilities add).
These basic axioms lead to a useful set of rules for finding probabilities of events.
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