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EMPIRICAL PROBABILITIES Among
the most important applications of probability are those situations where we
cannot list all possible outcomes. To this point, we have considered problems
in which the probabilities could be obtained from situations of equally likely
results. Because
some problems are so complicated for analysis, we can only estimate
probabilities from experience and observation. This is empirical probability.
In modern industry probability now plays an important role in many activities.
Quality control and reliability of a manufactured article have become extremely
important considerations in which probability is used. Table 91.Weather Forecast
Experience
has shown that empirical probabilities, if carefully determined on the basis of
adequate statistical samples, can be applied to large groups with the result
that probability and relative frequency are approximately equal. By adequate
samples we mean a large enough sample so that accidental runs of
"luck," both good and bad, cancel each other. With
enough trials, predicted results and actual results agree quite closely. On the
other hand, applying a probability ratio to a single individual event is
virtually meaningless. We
define relative frequency of success as follows: After N trials of an event
have been made, of which S trials are successes, the relative frequency of
success is ^{ } For
example, table 91 shows a small number of weather forecasts from April 1st to
April 10th. The actual weather on the dates is also given. Observe
that the forecasts on April 1, 3, 4, 6, 7, 8, and 10 were correct. We have
observed 10 outcomes. The event of a correct forecast has occurred 7 times.
Based on this information we might say that the probability for future
forecasts being true is 7/10. This number is the best estimate we can make from
the given information. In this case, since we have observed such a small number
of outcomes, we would be incorrect to say that the estimate of P is dependable.
A great many more cases should be used if we expect to make a good estimate of
the probability that a weather forecast will be accurate. A great many factors
affect the accuracy of a weather forecast. This example merely indicates
something about how successful a particular weather office has been in making
weather forecasts. Another example may be drawn from industry. Many thousands
of articles of a certain type are manufactured. The company selects 100 of
these articles at random and subjects them to very careful tests. In these
tests 98 of the articles are found to meet all measurement requirements and
perform satisfactorily. This suggests that 98/100 is a measure of the
reliability of the article. One might expect that about 98^{%}Io of all of the
articles manufactured by this process will be satisfactory. The probability
(measure of chance) that one of these articles will be satisfactory might be
said to be 0.98. This second example of empirical probability is different
from the first example in one very important respect. In the first example we
could list all of the possibilities, and in the second example we could not do
so. The selection of a sample and its size is a problem of statistics. Considered from another point of view, statistical
probability can be regarded as relative frequency. EXAMPLE: In a dart game, a player hit the bull's eye 3 times
out of 25 trials. What is the statistical probability that he will hit the
bull's eye on the next throw? SOLUTION: N=25 and S=3 hence
EXAMPLE: Using table 92, what is the probability that a
person 20 years old will live to be 50 years old? SOLUTION: Of 95,148 persons at age 20,
81,090 survived to age 50. Hence
EXAMPLE: How many times would a die be expected
to land with a 5 or 6 showing in 20 trials? Table
92.Mortality Table (Based on 100,000 Individuals 1 Year of Age)
SOLUTION: The probability of a 5 or 6 showing is ^{} The relative frequency is approximately equal to the
probability
Therefore, since
where
then rearranging and substituting, we find that
This says that the expected number of times a die would
land with a 5 or 6 showing in 20 trials is 6.67; that is, on the average a die
will land with a 5 or 6 showing 6.67 times per 20 trials. PRACTICE PROBLEMS: l. A construction crew consists of 6 electricians and 38
other workers. How many electricians would you expect to choose if you choose 1
person each day of a workweek for your helper? (Sunday will not be
considered part of the workweek.) 2. How many times would a tossed die be expected to turn
up a 3 or less in 30 tosses? 3. Using table 92, find the probability that a person
whose age is 30 will live to age 60. ANSWERS:
SUMMARY The following are the major topics covered in this
chapter: 1. Probability: Probability is the
ratio of the different number of ways a trial can succeed (or fail) to the
total number of ways in which it may result. 2. Probabilities of success and failure: If a trial must
result in any of n equally likely ways, and if s is the number of successful
ways and f is the number of failing ways, the probability
of success is
and the probability of failure is
3. Expectation: Expectation is the
average of the values you would get in conducting an experiment or trial
exactly the same way many times. 4. Numerical expectation: If the probability of success in
one trial is p, and k is the total number of trials, then kp is the expected
number of successes in the k trials or
5. Mathematical expectation: If, in the event of a
successful result, amount a is to be received, and p is the
probability of success of that event, then ap is the
mathematical expectation or
6. Independent events: Two or more events are independent
if the occurrence or nonoccurrence of one of the events has no effect on the
probability of occurrence of any of the others. 7. Dependent events: Two or more events are dependent
if the occurrence or nonoccurrence of one of the
events affects the probabilities of occurrence of any of the others. 8. Mutually exclusive events: Two or more events are
called mutually exclusive if the occurrence of any one of them precludes the
occurrence of any of the others. 9. Empirical probability: Empirical probability is an
estimated probability from experience and observation. 10. Relative frequency of success: After N trials of an
event have been made, of which S trials are successes, the relative frequency
of success is

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