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88 Introduction to probability
interested in calculating her mean (or average) weekly sales. This is easily
done by multiplying each sales level by its probability of occurrence and
summing the resulting products as shown below. The result is known
as an expected value.
No. of sets sold Probability No. of sets × probability
0 0.01 0.10
1 0.10 0.10
2 0.40 0.80
3 0.30 0.90
4 0.10 0.40
5 0.09 0.45
1.00 Expected sales = 2.65
It can be seen that an expected value is a weighted average with
each possible value of the uncertain quantity being weighted by its
probability of occurrence. The resulting figure represents the mean level
of sales which would be expected if we looked at the sales records over
a large number of weeks. Note that an expected value does not have
to coincide with an actual value in the distribution; it is obviously not
possible to sell 2.65 sets in a given week.
Although an expected value is most easily interpreted as ‘an average
value which will result if a process is repeated a large number of times’, as
we will see in the next chapter, we may wish to use expected values even
in unique situations. For example, suppose that a company purchases
its main raw material from a country which has just experienced a
military coup. As a result of the coup, it is thought that there is some
possibility that the price of the material will increase in the very near
future, and the company is therefore thinking of purchasing a large
supply of the material now. It estimates that there is a 0.7 probability
that the price will increase, in which case a saving of $350 000 will be
made, and a 0.3 probability that the price will fall because of other world
market conditions. In this case, purchasing early will have cost $200 000.
What is the expected saving of purchasing early? The calculations are
shown below:
Expected savings = (0.7 × $350 000) + (0.3 ×−$200 000)
= $185 000
Note that savings of $185 000 will not be achieved; the company will
either save $350 000 or lose $200 000. The figure is simply an average