For flights from a particular airport in January, there is a 30 percent chance of a flight being delayed because of icy weather. If a flight is delayed because of icy weather, there is a 10 percent chance the flight will also be delayed because of a mechanical problem. If a flight is not delayed because of icy weather, there is a 5 percent chance that it will be delayed because of a mechanical problem. If one flight is selected at random from the airport in January, what is the probability that the flight selected will have at least one of the two types of delays?

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The probability that the flight is delayed by icy weather and a mechanical problem:

$$ P(\text{Icy Weather})\cdot P(\text{Mechanical Problem}) = 0.30\cdot (0.10) = 0.03 $$

The probability that the flight is not delayed by icy weather and a mechanical problem:

$$ P(\text{Not Icy Weather}) \cdot P(\text{Mechanical Problem}) = (1-0.30)\cdot (0.05) = 0.035 $$

Adding the two:

$$ 0.03+0.035 = \boxed{0.0335} $$