Suppose I have $n^2\cdot \log^2 n+k$ pairs of boxes (each blue box has a corresponding red box), all independent and each having a coupon from the set $\{ 1,2,...,n\}$ with an equal probability (uniform distribution). By the coupon collector's, I can bound the probability that taking $n\cdot \log n+k$ blue boxes will result in having a coupon of each type.
Among the coupons, there is one type that is considered a winning coupon. To win the game, you need to find a winning coupon from the blue box and a winning coupon from its corresponding red box.
We assume all boxes are numbered, therefore each blue box has a corresponding red box. We also assume that we can distinguish between coupons from blue boxes and those of red boxes. Therefore, the only possibility of winning is having a winning coupon from a blue box and a winning coupon from its corresponding red box.
My idea is taking $n\cdot \log n+k$ blue boxes, knowing we have a winning coupon. Then it has some corresponding red box, might be winning and might not. If we take $n^2\cdot \log ^2n+k$ blue boxes, we will have $n\cdot \log n+k$ red boxes that correspond to a blue box with a winning coupon. And now we use the coupon collector on the set of red boxes...
Now, my question is: how to bound the probability of this procedure to succeed? From what I could tell, the probability is some constant to the $n\cdot \log n+k$ power, which is decreasing for large $n$.
Are there any ways to bound the probability of this procedure to succeed, in a better way? Or will my probability of success always be rapidly $\rightarrow 0$?