Consider this probability problem: An urn with 100 marbles. 90 are white, 10 are colored. Of the colored marbles three are red and seven are blue. When drawing out a single marble,
- P(Red) = .03),
- P(colored marble) = .1),
- P(Red|a colored marble) = .3
Now the events, "colored marble" and "Red Marble given a colored marble" are clearly not independent, but if you multiply P(colored) and P(Red| colored marble) you get P(Red). This seems to be a broad, frequently occurring exception to the rule that one can only multiply the probability of independent events. What am I missing?