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I was reading a riddle which says :

There are 10 boxes ,in each box there are ten apples which weigh 10g except for one box where the apples weigh 9g , find a way to find the special box when you can only weigh one time

The soultion is to take apples from each box with increasing order from 1 to 10 and then weighing them all together , then the difference will give the position of the box . The soultion is pretty creative but it made me wonder how can someone increase the probability just by changing some combinations. The probability was really low for one trial yet the soultion managed to make it 100% . I hope there is some mathematical clarification for this problem :)

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    I do not understand your question. "Increasing probability"... "the probability for one trial"... what trials? What probability? The solution to the problem you mention relies solely on properties of number theory and the fact that $\Bbb F_{11}$ is a field. It relies on the same mathematics as what is used for ISBN numbers and barcodes... – 2017-02-14
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    Make sure you study the answer below and understand that this problem has nothing to do with probability. @JMoravitz, you are right, but my guess is that this statement "$\Bbb F_{11}$ is a field" does not help the OP much :) – 2017-02-14

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By taking $k$ apples from box $k$, if it so happened that each apple had a weight of $10$ and none were weight $9$, we would have a total weight of

$$10\cdot \sum\limits_{k=1}^{10} k = 10\cdot 5\cdot 11 =550$$

If it so happens that box $n$ was the box that had apples of weight $9$ instead, then to correct the total weight from the above we need to subtract $n$ from it as instead of $10n$ being contributed from its box to the total weight it is instead $9n$ being contributed.

This brings the correct total weight to $550-n$

By subtracting this amount away from $550$, we get $550-(550-n)=n$ is the number of the box with the lighter apples.

This has nothing to do with probability, but is instead strictly a matter of algebra.