The probability that the economy will fall into a recession in any one year is 25%. Determine the probability that the economy will experience a recession in 1 out of 4 years and determine the expected number of recessions that will occur in 4 years. a. 25% probability of recession and 1 recession in the next four years. b. 25% probability of recession and 4 recessions in the next four years. c. 42% probability of recession and 2 recessions in the next four years. d. 42% probability of recession and 1 recession in the next four years. The answer is D. I get the expected number of recessions that will occur in 4 years =1, that’s pretty straight forward , its the 42% I just don’t completely understand.
4C1*(1/4)*(3/4)^3 = 42% you might want to read about binomial distribution here: http://en.wikipedia.org/wiki/Binomial_distribution
4!/(3!*1!)*0.25^1*0.75^3=42.19% And if you build up the binomial probability tree, it will get you 1 recession in 4 years:))
Binomial Distribution. There are only 2 states of nature here-Recession and No recession. So you get 4C1 (0.25^1)(1-0.25)^3=0.4218.
Think of it like this: The probability that the first year will be a recession and other three years will not is: .25 x .75 x .75 x .75. But the second year could be the recession or the third year or the fourth, so you have to use the combo formula to determine the number of different ways to pick 1 from 4 to account for these other possibilities. Obviously there are only 4 different ways to do this. So multiply 4 times the probability above and you get 42%.
OK so after reading these posts I was confused so I ended up doing the problem myself here are my calculation steps: Really the confusing part was knowing which formula to use 
Formula: Binominal Distribution n!/k!(n-k)! * p^k * (1-p)^n-k 1) n!/k!(n-k)! 4!/1!(4-1)! ----> 4!/1!(3!) -----> = 4 2) p^k .25^1 ----> = .25 3) (1-p)^n-k (1-.25)^4-1 ----> (.75)^3 ----> =.4219 4) 4*.25*.4219 = .4219 REALLY you could have just .75^3 and wallah gotten the answer .4219 but i don’t recommend that since it prob. doesn’t work all the time…only in this case.
Yuck. This question is assuming independence, which I see no reason to believe is true.
whodey Wrote: ------------------------------------------------------- > The probability that the economy will fall into a > recession in any one year is 25%. Determine the > probability that the economy will experience a > recession in 1 out of 4 years and determine the > expected number of recessions that will occur in 4 > years. > > a. 25% probability of recession and 1 recession in > the next four years. > b. 25% probability of recession and 4 recessions > in the next four years. > c. 42% probability of recession and 2 recessions > in the next four years. > d. 42% probability of recession and 1 recession in > the next four years. > > The answer is D. > > I get the expected number of recessions that will > occur in 4 years =1, that’s pretty straight > forward , its the 42% I just don’t completely > understand. D Lets list the favorable scenario’s 1/4*3/4*3/4*3/4 = 27/64*4 No realize we have 4C1 ways to arrange this possible scenario. Thus we have 27/64 or ~.42
lognormal Wrote: ------------------------------------------------------- > Yuck. This question is assuming independence, > which I see no reason to believe is true. thats the case in real life but in this case its assumed that all 4 years are independent of each other. Also this is case of binomial probability so just use that formula
I almost went rusty remembering the binomial probability formula. illcaprice had a great explanation for this sort of problem. Your given four years. One out of those four can be a recession. That’s four possible scenarios (recession in year 1, year 2, year 3, or year 4). Order does not matter, so you can solve using the combination function on your calculator. Specifically: [4], [2nd] [+], [1], [=] The answer, 4, gives you the first component. The second component is the probability of recession raised to it’s expected frequency: p^x That gives us 0.25^1 = 0.25 The third component is the probability of not having a recession, raised to it’s expected frequency: (1-p)^(n-x) That gives us (1 - 0.25)^(4-1) = 0.42 (rounded) By multiplying the last two components, we get the probability of a recession in a specific year, while the other years don’t experience a recession. For example, the probability of a recession only in year 2 our of all 4 years is 0.105. In the grand scheme of things, we’re only concerned with the probability relating to the bigger picture, which is 1 recession out of a 4 year period. That’s where the first component, the combination, comes in–it gives us the total number of scenarios. Multiplying that with the the other two components gives you 0.42. You can change it up a bit, too. For instance, ask yourself, what’s the probability of 2 recessions in a 4 year period? Or what is the probability of experiencing at least 2 recessions in a 4 year period?
Destination.CFA Wrote: ------------------------------------------------------- > lognormal Wrote: > -------------------------------------------------- > ----- > > Yuck. This question is assuming independence, > > which I see no reason to believe is true. > > thats the case in real life but in this case its > assumed that all 4 years are independent of each > other. > > Also this is case of binomial probability > so just use that formula My problem is that the question is implicitly assuming independence without explicitly stating such. The severity of that problem is compounded by the fact that in real life independence doesn’t hold here.