In the study of a storage dam design, it is assumed thatquantities can be measured sufficiently accurately in units of ¼ ofthe dam’s capacity. It is known from past studies that at thebeginning of the first (fiscal) year the dam will be either full, ¾full, ½ full, or ¼ full, with probabilities 1/3, 1/3, 1/6, and 1/6,respectively. During each year water is released. The amountreleased is ½ the capacity if at least this much is available; itis all that remains if this is less than ½ the capacity. Afterrelease, the inflow from the surrounding watershed is obtained. Itis either ½ or ¼ of the dam’s capacity with probabilities 2/3 and1/3, respectively. Inflow causing a total in excess of the capacityis spilled. Assuming independence of annual inflows, what is theprobability distribution of the total amount of water at thebeginning of the third year? Use total probability theorem.
