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74 Introduction to probability
practical situations (e.g. the simultaneous product launch above) the
outcomes will not be equally likely and therefore the usefulness of this
approach is limited.
The relative frequency approach
In the relative frequency approach the probability of an event occurring
is regarded as the proportion of times that the event occurs in the long run
if stable conditions apply. This probability can be estimated by repeating
an experiment a large number of times or by gathering relevant data and
determining the frequency with which the event of interest has occurred
in the past. For example, a quality control inspector at a factory might
test 250 light bulbs and find that only eight are defective. This would
suggest that the probability of a bulb being defective is 8/250 (or 0.032).
The reliability of the inspector’s probability estimate would improve as
he gathered more data: an estimate based on a sample of 10 bulbs would
be less reliable than one based on the sample of 250. Of course, the
estimate is only valid if manufacturing conditions remain unchanged.
Similarly, if the publisher of a weekly magazine found that circulation
had exceeded the break-even level in 35 out of the past 60 weeks then
he might estimate that the probability of sales exceeding the break-even
level next week is 35/60 (or 0.583). Clearly, for this probability estimate
to be reliable the same market conditions would have to apply to every
week under consideration; if there is a trend or seasonal pattern in sales
it would not be reliable.
This raises the problem of specifying a suitable reference class. For
example, suppose that we wish to determine the probability that Mary,
a 40-year-old unemployed computer programmer, will find a job within
the next 12 months. By looking at recent past records we might find that
30% of unemployed people found jobs within a year and hence estimate
that the probability is 0.3. However, perhaps we should only look at
those records relating to unemployed female computer programmers of
Mary’s age and living in Mary’s region of the country, or perhaps we
should go even further and only look at people with similar qualifications
and take into account the fact that Mary has a record of ill health. Clearly,
if the data we used were made too specific it is likely that we would
find that the only relevant record we had related to Mary herself. It can
be seen that there is a conflict between the desirability of having a large
set of past data and the need to make sure that the data relate closely to
the event under consideration. Judgment is therefore required to strike
a balance between these two considerations.