Use the following information to answer question. A manufacturer is planning its production for the next 6 months. It has to decide how much of an important ingredient to keep in inventory. The demand for the ingredient may be low, medium, or high. The manufacturer is considering holding either a low or a high amount of inventory. The possible payoffs for all the combinations of inventory holding and demand are shown here.

An economics consultant predicts that the demand for the ingredient will be low in the next 6 months. In the past few years, this economist has provided forecasting about the demand for the ingredient. The track record of the consultant is summarized by the following conditional probability distribution:

Assume that this time, the economist predicts a low demand for the future. Find the posterior distribution of S₁, of S₂, and of S₃.

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Inventory holding Low A High A Probability Demand Low S 200 100 .3 Medium S 300 200 .2 High S3 300 500 .5