Independent and dependent events, and probability from area
Two events are independent when one happening does not change the probability of the other, so P(A and B) = P(A) x P(B). They are dependent when it does, so P(A and B) = P(A) x P(B given A). Drawing with replacement keeps draws independent, while drawing without replacement makes them dependent. From a bag with 3 red and 2 blue marbles, the probability of two reds with replacement is (3/5)(3/5) = 9/25, but without replacement it is (3/5)(2/4) = 3/10, because one red is gone before the second draw. Probability can also come from area: if a dart lands at a random point on a 10-by-10 board, the chance it lands inside a circle of radius 2 drawn on the board is the circle's area over the board's area, (pi x 2^2)/100, about 0.13.
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