Question

In the book Making Hard Decisions: An Introduction to Decision Analysis, 2nd ed., Robert T. Clemen presents an example in which an investor wishes to choose between investing money in (1) a high-risk stock, (2) a low-risk stock, or (3) a savings account. The payoffs received from the two stocks will depend on the behavior of the stock market-that is, whether the market goes up, stays the same, or goes down over the investment period. In addition, in order to obtain more information about the market behavior that might be anticipated during the investment period, the investor can hire an economist as a consultant who will predict the future market behavior. The results of the consultation will be one of the following three possibilities: (1) "economist says up," (2) "economist says flat" (the same), or (3) "economist says down." The conditional probabilities that express the ability of the economist to accurately forecast market behavior are given in the following table: Economist's Prediction "Economist says up" "Economist says flat" "Economist says down" True Market State Up Flat Down 0.80 0.10 0.10 0.10 0.65 0.25 0.10 0.15 0.75 For instance, using this table we see that Pl"economist says up" I market up) = .80. Figure 19.7 gives an incomplete decision tree for the investor's situation. Notice that this decision tree gives all relevant payoffs and also gives the prior probabilities of up, flat, and down, which are, respectively, 55, 20, and .25. Using the information provided here, set up probability revision tables to (a) Find the probability that the "economist says up" and find the posterior probabilities of market up, market flat, and market down given that the "economist says up." (Round intermediate calculations and final answers to 4 decimal places.) P("economist says up") = Up Flat Down Posterior Probabilities: (b) Find the probability that the "economist says flat," and find the posterior probabilities of market up, market flat, and market down given that the "economist says flat." (Round intermediate calculations and final answers to 4 decimal places.) P("economist says flat") Up Flat Down Posterior Probabilities:

Answer 1

probabilities: "Says up 0.8 O, 1 0,1 0.65 0.25 0.19 -0.75 Conditional True Market State Economist's Pred Up Flat Down says flat" Says says down Prior probs a PC True Market = 'up) = 0.55 PCGM = sat') = 0.20 PLTM = doon') : 0.25, (E = 'up") P CF-up) = PCE-up 174, up] PC7M sup) + PC cupla doon) & PLE=down 17.M= flat) PCT. M flat) (2) PL Economist says up") => P (E=

this comes from on the sum rule P(A) = P(AIB) PUB] + PLAIBO) PUBC). Do, PCE zup) = 0.8 xoss + o.lx0.25 tot 20:20 6.4 Pro prob. Baye's rule, Posterion PLTM "up lerup from PLBLAD = PCAIB) PCB) PIA BPCB] + PCMIBOYPUB) (•) PeTM = 'up'! Ez 'up'? - PLE = 'uplep PL 7,14:="p? PLE='up') 0.8 x 8.552 0.592 0.2134 0.489 c) pc flat E= 'up") = P CE Eup leflet IP Gflat) PLE=up) (i) P(down 1E= = "up) Ool 0 .20 -0.0412 0.485 0.1% 0.25 = 0.0515 00489

PLE=up) (%) Pl down 1E= E = "up) Oolx 0 20 -0.0412 0.485 o.lx0.25 = 0.0515 00489 (6 PL Economist says "flat") P (E=flat (flat) peplat) + PLE-flat I up) plup) + Pies flat Idown) p down) oobs Xo.20 +0.1X a.sst o.as xoias 0.2475 O P Cup (E = flat) = PLE=flat l ep. 91 Cup) 0,1 x 0.55 PLE=flat ) 0.2222 0.2475 Plflat lE = flat ) = 0.65*0.20.0.5252 (iv) Pldocon 1E = flay ) 0125 xords - 0.2525 0.2475 0.2475

docan © PCE= "locor') - PCE-leon Icoon ) M Cleven ) + PCE=op lupTplus APLE=downlflat) 84727) 0.75 to.as to. 1x o.rs + 0,15x 0.20 0,2725 () P Pleple= ple = down) PLE=down (up) plep) PLE=foon') 0.1X0.59 = 0,2018 D. 2729 PC flat IE - Lowon ) = 0.1540.20 oolloo 0.2725 (1/1) p Codacon IE= down) = 0.25x 0,25 -0.6880 . 0.2725