The authors introduce a general model for physical sources of weak randomness. Loosely speaking, they view physical sources as devices which output strings according to probability distributions in which no single string is too probable. The main question addressed is whether it is possible to extract almost unbaised random bits from such 'probability bounded' sources. It is shown that most of the functions can be used to extract almost unbiased and independent bits from the output of any two independent probability-bounded sources. The number of extractable bits is within a constant factor of the information-theoretic bound. Further connections are established between communication complexity and the problem discussed above. This makes it possible to show that most Boolean functions have linear communication complexity in a very strong sense.