I'm trying to prepare for a test and going over some practice problems I just found with no solutions, and they've basically led me to believe I'm completely lost. The section we're working on is joint probability, while the entire section in the book focused on problems with pdf's given as formulas and nothing else. I think I understand that stuff, but I think these words problems are mixing in some of the older content and for some reason I'm just having a huge mental block and struggling to bring it all together.
I know that the probability of an even number of dots is 3/6 and the probability of a 5 is 1/6, but I'm struggling at understanding how to apply this to 12 rolls with both variables included. I'm looking through the book for examples but I'm not seeing anything similar and seem to be hitting a mental block here.
On this one I started out just trying to calculate f(x,y) and setting up an interval. I came up with f(x,y)=0.25(e^(-x/4)*e^(-y/4)) and then solved over the intervals 1-2 for x and 2-3 for y. I came up with the solution of F(1<x<2,2<y<3)=0.0925 and while I'm not even sure that part is right, I don't know where to go from here. My belief is that the 0.0925 probability is the chance for one item to fail during each of these intervals, but how do I apply this to failing 3/9 attempts on x and 4/9 attempts on y?
1) A die is rolled 12 times. a) Compute the probability that an even number of dots occurs 5 times and 5 dots occurs once in the 12 rolls.
Suppose that X=time to failure for a component has an exponential distribution with lambda =.25. Suppose that 9 of the components are selected and their failure times noted. Compute the probability that 3 of the components fail between times 1 and 2 and 4 of the components fail between times 2 and 3. Assume that the failure times are independent.