A certain financial market experiences growth (G) 20% of the time, stagnation (S) 70% of the time, and recession (R) 10% of the time. A consulting firm is hired to forcast this market. In order to assess the reliability of this firm's forcasts, we investigate past performance of this consulting firm. Records show that whenever there was growth, the firm had correctly forcasted it 80% of the time. The firms accuracy was the same for periods of stagnation and recession, that is, it forcasted correctly 80% of the time and made each of the other two errorous forecasts 10% of the time. The tree for this two stage experiment has three branchings at each stage. Suppose that now the firm forcasts growth. Given the forecast and the past performance, find the probability of a.) growth, b.) stagnation, and c.) recession.
I drew the tree diagram and got the following values once the multiplication was completed: 0.16, 0.02, 0.02, 0.07, 0.56, 0.07, 0.01, 0.01, 0.02. The only thing I don't understand is what my probability equations should look like.
Pr(Growth|Forcast and Performance) = ??
Pr(Stagnation|Forcast and Performance) = ??
Pr(Recession|Forcast and Performance) = ??
How do I break up these equations (that is if they are even right), so that I can input numbers to get the probability values for a.) , b.) , and c.).