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# utf-8__STOCHPROC_001 - P ROBLEMS FOR THE MIDDLE TERM EXAM[1...

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P ROBLEMS FOR THE MIDDLE TERM EXAM [1] Given three sets A, B, C . Use the Venn diagram to illustrate the following sets: A B C , A B B ( C \ ( A B )) . [2] Let a card be selected from two ordinary packs of 52 cards. Denote A = { the card is Diamonds or clubs } , B = { the card is a red suit one } , C = { the card is not a Heart Jack } . Compute the following probabilities: P ( A ) , P ( B C ) , P ( C | B ) [3] EX 1( 2 points) : What are a sample space, events an their probilitiy? Formulate axioms of probability ? Give an example ? EX 2 ( 2 points) : What is the conditional probability of an event A given E . When three events A i , i = 1 , 2 , 3 are independent ? Give an example of a sequence of dependent ( non-independent) events A i , i 3 such that each pair of its membrs are independent. EX 3 (2 points) : A pair of fair dice is tossed. We obtain the finite equiprobable Ω consisting of the 36 ordered pairs of numbers between 1 and 6 Ω = { (1 , 1) , (1 , 2) , · · · , (6 , 6) } . Let X assign to each point ( a, b ) in Ω the minimum of its numbers, i.e X ( a, b ) = min( a, b ). What is the image set of random variable

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