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Unformatted text preview: ceptual problem that was posed in
Section 1.1. Suppose we have a probability model of some realworld experiment involving
randomness in the sense expressed there. When the realworld experiment being modeled is
performed, there is an outcome, which presumably is one of the outcomes of the probability
model, but there is no observable probability.
It appears to be intuitively natural, for experiments that can be carried out repeatedly
under essentially the same conditions, to associate the probability of a given event with 1.5. RELATION OF PROBABILITY MODELS TO THE REAL WORLD 43 the relative frequency of that event over many repetitions. We now have the background
to understand this approach. We ﬁrst look at relative frequencies within the probability
model, and then within the real world. 1.5.1 Relative frequencies in a probability model We have seen that for any probability model, an extended probability model exists for n
IID idealized experiments of the original model. For any event A in the original model,
the indicator function IA is a random variable, and the relative frequency of A over n IID
experiments is the sample average of n IID rv’s with the distribution of IA . From the weak
law of large numbers, this relative frequency converges in probability to E [IA ] = Pr {A}. By
taking the limit n → 1, the strong law of large numbers says that the relative frequency
of A converges with probability 1 to Pr {A}.
In plain English, this says that for large n, the relative frequency of an event (in the nrepetition IID model) is essentially the same as the probability of that event. The word
essential ly is carrying a great deal of hidden baggage. For the weak law, for any ≤, δ > 0,
the relative frequency is within some ≤ of Pr {A} with a conﬁdence level 1 − δ whenever n is
suﬃciently large. For the strong law, the ≤ and δ are avoided, but only by looking directly
at the limit n → 1. 1.5.2 Relative frequencies in the real world In trying to sort out if and when the laws of large numbers have much to do with realworld
experiments, we should ignore the mathematical details for the moment and agree that for
large n, the relative frequency of an event A over n IID trials of an idealized experiment
is essentially Pr {A}. We can certainly visualize a realworld experiment that has the same
set of possible outcomes as the idealized experiment and we can visualize evaluating the
relative frequency of A over n repetitions with large n. If that realworld relative frequency
is essentially equal to Pr {A}, and this is true for the various events A of greatest interest,
then it is reasonable to hypothesize that the idealized experiment is a reasonable model for
the realworld experiment, at least so far as those given events of interest are concerned.
One problem with this comparison of relative frequencies is that we have carefully speciﬁed
a model for n IID repetitions of the idealized experiment, but have said nothing about how
the realworld experiments are repeated. The IID idealized experiments specify that the
conditional probability of A at one trial is the same no matter what the results of the other
trials are. Intuitively, we would then try to isolate the n realworld trials so they don’t aﬀect
each other, but this is a little vague. The following examples help explain this problem and
several others in comparing idealized and realworld relative frequenices.
Example 1.5.1. Coin tossing: Tossing coins is widely used as a way to choose the ﬁrst
player in other games, and is also sometimes used as a primitive form of gambling. Its
importance, however, and the reason for its frequent use, is its simplicity. When tossing
a coin, we would argue from the symmetry between the two sides of the coin that each
should be equally probable (since any procedure for evaluating the probability of one side 44 CHAPTER 1. INTRODUCTION AND REVIEW OF PROBABILITY should apply equally to the other). Thus since H and T are the only outcomes (the remote
possibility of the coin balancing on its edge is omitted from the model), the reasonable and
universally accepted model for coin tossing is that H and T each have probability 1/2.
On the other hand, the two sides of a coin are embossed in diﬀerent ways, so that the mass
is not uniformly distributed. Also the two sides do not behave in quite the same way when
bouncing oﬀ a surface. Each denomination of each currency behaves slightly diﬀerently in
this respect. Thus, not only do coins violate symmetry in small ways, but diﬀerent coins
violate it in diﬀerent ways.
How do we test whether this eﬀect is signiﬁcant? If we assume for the moment that successive tosses of the coin are wellmodeled by the idealized experiment of n IID trials, we can
essentially ﬁnd the probability of H for a particular coin as the relative frequency of H in a
suﬃciently large number of independent tosses of that coin. This gives us slightly diﬀerent
relative frequencies for diﬀerent coins, and thus slightly diﬀerent probability models for
diﬀerent coins. If we want a generic model, we might randomly choose coi...
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This note was uploaded on 09/27/2010 for the course EE 229 taught by Professor R.srikant during the Spring '09 term at University of Illinois, Urbana Champaign.
 Spring '09
 R.Srikant

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