If I win the bet I get 200 dollars and my winning probability is 0.1 and if I lose I give 20 dollars. Should I take the bet?
It depends on many factors. First - are you taking the bet once or as long as you want?
- If you take it just once then the answer is no, as there is much higher probability of losing money then winning(even though the expected value is positive)
- If you can take this bet for as long as you want, it is a good bet, as with expected value of income equal to 2, we could expect that if we play long enough - we will gain more and more money
In general the problem here is weighting the expected outcome (expected value) and the risk (variance) of the process. In "real life", where all resources are limited, the risk factor may be much bigger then the one coming from expected outcome (which is just a theoretical entity). As such there is no good answer for "real world" bet, only for theoretical betting games, where there exists infinity