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Statistics

Notes Solution.

(i) The minimum payoffs for various actions are :
Action 1 = - 5
Action 2 = - 2

Action 3 = 2
Action 4 = - 3
Since the payoff for action 3 is maximum, therefore, A is optimal on the basis of maximin
3
criterion.
1
(ii) Since there are 5 equally likely events, the probability of each of them would be .
5
+
4 0 5 3 6 8
-
+
+
Thus, the EMV of action 1, i.e., EMV = = = 1.6
1
5 5
20 19 17
Similarly, EMV = = 4.0 , EMV = = 3.8 and EMV = = 3.4
2
3
4
5 5 5
Thus, action 2 is optimal.
8.3 Use of Posterior Probabilities in Decision Making
The probability values of various states of nature, discussed so far, were prior probabilities.
Such probabilities are either computed from the past data or assigned subjectively. It is possible
to revise these probabilities in the light of current information available by using the Bayes'
Theorem. The revised probabilities are known as posterior probabilities.

Example 22: A manufacturer of detergent soap must determine whether or not to expand
his productive capacity. His profit per month, however, depends upon the potential demand for
his product which may turn out to be high or low. His payoff matrix is given below :

Do not Expand Expand
High Demand Rs 5,000 Rs 7,500
Low Demand Rs 5,000 Rs 2,100

On the basis of past experience, he has estimated the probability that demand for his product
being high in future is only 0.4

Before taking a decision, he also conducts a market survey. From the past experience he knows
that when the demand has been high, such a survey had predicted it correctly only 60% of the
times and when the demand has been low, the survey predicted it correctly only 80% of the
times.
If the current survey predicts that the demand of his product is going to be high in future,
determine whether the manufacturer should increase his production capacity or not? What
would have been his decision in the absence of survey?

Solution.
Let H be the event that the demand will be high. Therefore,
P(H) = 0.4 and P(H) = 0.6
Note that H and H are the only two states of nature.

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