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Unformatted text preview: 6.042/18.062J Mathematics for Computer Science April 26, 2005 Srini Devadas and Eric Lehman Lecture Notes Independence 1 Independent Events Suppose that we ip two fair coins simultaneously on opposite sides of a room. Intu- itively, the way one coin lands does not affect the way the other coin lands. The mathe- matical concept that captures this intuition is called independence . In particular, events A and B are independent if and only if: Pr ( A B ) = Pr ( A ) Pr ( B ) Generally, independence is something you assume in modeling a phenomenon or wish you could realistically assume. Many useful probability formulas only hold if certain events are independent, so a dash of independence can greatly simplify the analysis of a system. 1.1 Examples Lets return to the experiment of ipping two fair coins. Let A be the event that the first coin comes up heads, and let B be the event that the second coin is heads. If we assume that A and B are independent, then the probability that both coins come up heads is: Pr ( A B ) = Pr ( A ) Pr ( B ) 1 1 = 2 2 1 = 4 On the other hand, let C be the event that tomorrow is cloudy and R be the event that tomorrow is rainy. Perhaps Pr ( C ) = 1 / 5 and Pr ( R ) = 1 / 10 around here. If these events were independent, then we could conclude that the probability of a rainy, cloudy day was quite small: Pr ( R C ) = Pr ( R ) Pr ( C ) 1 1 = 5 10 1 = 50 Unfortunately, these events are definitely not independent; in particular, every rainy day is cloudy. Thus, the probability of a rainy, cloudy day is actually 1 / 10 . 2 Independence 1.2 Working with Independence There is another way to think about independence that you may find more intuitive. According to the definition, events A and B are independent if and only if: Pr ( A B ) = Pr ( A ) Pr ( B ) . The equation on the left always holds if Pr ( B ) = 0 . Otherwise, we can divide both sides by Pr ( B ) and use the definition of conditional probability to obtain an alternative definition of independence: Pr ( A | B ) = Pr ( A ) or Pr ( B ) = 0 This equation says that events A and B are independent if the probability of A is unaf- fected by the fact that B happens. In these terms, the two coin tosses of the previous section were independent, because the probability that one coin comes up heads is un- affected by the fact that the other came up heads. Turning to our other example, the probability of clouds in the sky is strongly affected by the fact that it is raining. So, as we noted before, these events are not independent. 1.3 Some Intuition Suppose that A and B are disjoint events, as shown in the figure below. A B Are these events independent? Lets check. On one hand, we know Pr ( A B ) = 0 because A B contains no outcomes. On the other hand, we have Pr ( A ) Pr ( B ) > 0 except in degenerate cases where A or B has zero probability. Thus, disjointness and inde- pendence are very different ideas ....
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