How to sum two chances?

How to calculate cumulative chance from 2 different sources that are deciding on 1 result.

For example, there is a 1/4 chance from bonus and 1/10 chance in general. When we put those 2 together what do we get?

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It depends on whether the two 'chances' are independent. For example, suppose I know that there is a $1/2$ chance that it will rain tomorrow, a $1/2$ chance that it will snow tomorrow, and a $1/4$ chance that it will both rain and snow tomorrow. What is the probability that it will either rain or snow (or perhaps both) tomorrow?
If we simply add $1/2 + 1/2$, we are overcounting. Some of the chances that it will rain are tied to the chances that it will snow, as the probability of both happening is $1/4$. So we would say that the chance of rain or snow tomorrow is $1/2 + 1/2 - 1/4$.
This is analogous to the idea that maybe there are 2 boys in a chemistry class, 3 boys in an English class, and we happen to know that there is 1 boy in both. How many boys are in the chemistry and English classes? There are almost $2 + 3$, but we counted 1 boy twice. So there are $2 + 3 - 1$.
All together, this means that I don't know the exact answer to your question. It will be somewhere between $25\%$ and $35\%$. $%Any airplane fans out there? All together now...$