3. An economist studying the value of analysts' forecasts of corporate earnings collects 1000 forecasts of earnings increase, decrease, or constancy. The results on the sample can be broken down as follows (F=forecast, O=actual outcome, I=increased, D=decreased, C=approximately constant):

F:I O:I - 210

F:I O:C - 119

F:I O:D - 65

F:C O:I - 82

F:C O:C - 166

F:C O:D - 72

F:D O:I - 76

F:D O:C - 84

F:D O:D - 126

a ) Find the probability that a forecast of lower earnings will prove to be correct.

b ) Find the probability that a forecast of higher earnings will prove to be wrong.

Now the investigator considers a further sample of an analyst's picks, that is, stocks expected by the analysts to outperform the stock market index. The returns on a sample of stocks are divided into those doing better (25%), about the same (50%) and worse (25%) than the index. Of those doing better, 40% were rated to outperform by the analyst; so were 20% of those doing about the same, and 10% of those doing worse. What is the probability that a stock rated to outperform did in fact perform better?