Each box contains a pen with one of four colors, equally likely. Each box is independent.

After opening N (N=4) boxes, what is probability you have a full set (all 4 colors)?

My daughter came up with the following solution which I verified with a quick Matlab run.

P(set)=P(any color in box 1)*P(different color in box 2)*P(different color in box 3)*P(different color in box 4)

P(set)=1*.75*.5*.25

P(set)=.094

I tried a more formal method (because the next problem requires a higher value of N) and got a different answer.

P(set)=P(at least one blue)*P(at least one red|at least one blue)*P(at least one yellow|at least one red & blue)*P(at least one green|at least one red & blue & yellow)

P(at least one blue)=[1-P(no blue in any box)]=[1-.75^4]=.684

P(at least one red|at least one blue)=[1-P(no red in 3 boxes)]=[1-.75^3]=.578

P(at least one yellow|at least one red & blue)=[1-P(no yellow in 2 boxes)]=[1-.75^2]=.438

P(at least one green|at least one red & blue & yellow)=[1-P(no green in last box)]=[1-.75]=.25

So, P(set)=.684*.578*.438*.25=.043

I know the first answer is right. I don't know why the second method doesn't work. Can someone please point out my mistake?