Berkeley STAT 134 - Lecture Notes (10 pages)

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Lecture Notes



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Lecture Notes

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Lecture Notes


Pages:
10
School:
University of California, Berkeley
Course:
Stat 134 - Concepts of Probability
Concepts of Probability Documents

Unformatted text preview:

Events A and B are independent if knowing whether A occured does not change the probability of B Mathematically can say in two equivalent ways P B A P B P A and B P B A P B P A Important to distinguish independence from mutually exclusive which would say B A is empty cannot happen Example Deal 2 cards from deck A first card is Ace C second card is Ace P C A P C So A and C are dependent 4 52 last 3 51 class Example Throw 2 dice A first die lands 1 B second die shows larger number than first die C both dice show same number P B A 5 6 P B 15 36 by counting so A and B dependent P C A 1 6 P C 6 36 1 6 so A and C independent Note 1 here B and C are mutually exclusive Note 2 writing B second die shows smaller number than first die we have P B P B by symmetry P B B P C c 1 P C giving a non counting argument that P B 5 12 5 6 Example Deal 1 card from deck A card is Ace S card is Spade P A 4 52 P S 13 52 P A S 1 52 Here P A S P A P S so independent Conceptual point a In a fully specified math model two events are either dependent or independent can be checked by calculation b Often we use independence as an assumption in making a model For instance we assume that di erent die throws give independent results Most probability models one encounters in engineering or science have some assumption of bottom level independence but one needs to be careful about which other events within the model are independent silly Example Throw 2 dice If sum is at least 7 I show you the dice if not I don t A I show you first die lands 1 B I show you second die lands 1 P A 1 36 P B 1 36 P A B 0 so A and B dependent Conceptual point This illustrates a subtle point being told by a truthful person that A happened is not for probability statistics purposes exactly the same as knowing A happened car accident example Systems of components Will show logic diagrams system works if there is some path left to right which passes only though working components Assume components work fail independently P Ci



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