There is a 45% probability of rising interest rates with no inflation and a 45+65%=110% probability of rising interest rates with inflation. What is the probability of inflation when interest rates are rising, given a 60% chance of inflation. (a) 0.34 (b) 0.68 © 0.57 (d) 0.60
Hi, this question makes my brain hurt. Is the wording regarding the 110% probability correct? Ie, 110% probability doesn’t make any sense, are you sure the 110% doesn’t refer to some kind of rate or return instead?
Stratus, This is one of Stalla’s homework questions. I have no idea what that 110% probability means, but her is the answer they came up with: Choice “b” is correct. This question requires updating our probability of inflation during a rising interest rate environment with new information about the 60% probability of inflation. There is no information about whether interest rates will be rising or falling. Let I denote inflation, N denote no inflation, R denote rising interest rates and F denote falling interest rates. The probability of inflation when interest rates are rising, given the updated 60% probability of inflation is: P® = P(R|I) P(I) + P(R|N) = 0.65X0.60 + 0.45X0.40) =0.57 P(I|R) = P(IR)/P(® = P(I) P(R|I) / P® = (0.6)(0.65) / 0.57 = 0.6842 Choice “a” is incorrect. This choice incorrectly multiplies the probability of rising rates by the probability of inflation, rather than calculating the probability of inflation given rising interest rates. Choice “c” is incorrect. This is the probability of rising interest rates rather than the updated probability of inflation with rising interest rates. Choice “d” is incorrect. This is the probability of inflation rather than the updated probability of inflation when interest rates are rising.
This is a VERY poorly worded problem. Their solution uses standard Bayesian inference, but their definitions of the probabilities are horrifically stated. There is also mistake in their solution (unless you typed it wrong). P(I) is prob of inflation = 0.6. Therefore P(N) = prob of no inflation = 0.4 If there is no inflation, probability of rising rates is P(R|N)=0.45 If there is inflation, probability of rising rates is P(R|I)=0.65 We are asked to find P(I|R), or the probability of inflation given that rates are rising. That is how the problem should have been worded (without the P(N)'s and other algebra). Total probability of rising rates P®=P(R|I)P(I) + P(R|N)P(N) = 0.57 (when you plug in the numbers above. You missed that P(N) when you typed it out). Bayes theorem gives P(I|R)=p(R|I)p(I)/p® = 0.68 Answer is B.
Hey guys I had reported an error on this one, the very first time I saw it, and it was corrected subsequently. Please refer to the errata on Stalla. Date: 1/5/07 Session: 2 Pg: 101 Question #9 should read as: “There is a 45% probability of rising interest rates with no inflation and a 65% probability of rising interest rates with inflation.”