Q:

What are the chances of at least two out of three with different odds for each? 

Let's say you make three bets... first one has 30% chance of winning, second has 40%, and last has 50%.   What are the odds that you win at least two of them?   How is that calculated? 
So far I know that I can't just add them together.   I can't multiply them together.   I tried manually doing each pair (.3*.4 , .4*.5, .3*.5) to calculate the odds of those pairs both winning... but I'm not sure how to add them together. 

A:

Hint: Let $A_i$ the probability that you win bet $i$ with the following probabilities:
$P(A_1)=0.3, P(A_2)=0.4, P(A_3)=0.5$ and $P(\overline A_i)=1-P(A_i)$
Then the probability to win at least 2 bets is
$P(A_1)\cdot P(A_2) \cdot P(\overline A_3)+P(A_1)\cdot P(\overline A_2) \cdot P( A_3)+P(\overline A_1)\cdot P( A_2) \cdot P( A_3)$ $+P( A_1)\cdot P( A_2) \cdot P( A_3)$

