Weighing the heavier one!

Question

There are 5 pairs of balls which are kept in 5 different boxes (i.e. each box has 2 identical balls). of the boxes both the balls weigh 9g each. In the remaining 4 boxes all balls weigh 8g each. You have a weighing machine with two pans . If you put some balls on the left pan and some on the right, its reading will show you the (value of the weight on the left pan - value of weight on the right pan). Find the minimum number of times the weighing machine needs to be used in order to identify the balls which are heavier. The difference can be 2 or 1 or 0 0 has two cases Please help

Answer 1

Apparently it is impossible to weigh only one time since even if you remove one box temporarily and weigh the four remaining boxes you still have ambiguity (the other cases don't give any extra information such as removing two boxes or more). Therefore there need to be two weighing 2 times at worst case. First step, put aside a box and weigh two remaining pair boxes in the weighing machine. If there were equal the aside box is that we're looking for (This is the best case and happens with probability of $\frac{1}{5}$). Otherwise among those pick up the pair with higher weight and do weighing among two remaining boxes to reveal the true one (This is the worst case and happens with probability $\frac{4}{5}$). So there is at least 1 and at most 2 times weighing needed and the average is $1*\frac{1}{5}+2*\frac{4}{5}=\frac{9}{5}$