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# Homework 2 - 1 1 6 6 6 and D =(3 3 3 3 3 6 We reserve the...

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550.420 Introduction to Probability Fall 2011 Homework 2 Due Thursday September 15 1. (2 points) Let E , F , and G be three events. Determine which of the following statements are correct and which are incorrect. Justify your answers. (a) ( E - EF ) F = E F. (b) F c G E c G = G ( F E ) c . 2. (2 points) If P ( A ) = 0 . 7, P ( B ) = 0 . 6, P ( C ) = 0 . 3, and P ( AC ) = 0 . 2, find the maximum and minimum possible values of P ( A BC ). 3. (2 points) Which of the following statements are true? If a statement is true, prove it. If it is false, give a counterexample. (a) If P ( A ) + P ( B ) + P ( C ) = 1, then the events A , B , and C are mutually exclusive. (b) If P ( A B C ) = 1, then A , B , and C are mutually exclusive events. 4. (4 points) Abnormal Dice: Consider a game played with three dice. Each of the dice is a cube with 21 spots - the same as an ordinary die. However, the spots are distributed differently than on an ordinary die. One had four spots on each of five faces, and one spot on the other face - we will denote it by B = (1 , 4 , 4 , 4 , 4 , 4). The other two die are C = (1
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Unformatted text preview: , 1 , 1 , 6 , 6 , 6) and D = (3 , 3 , 3 , 3 , 3 , 6). We reserve the name A for the standard die, i.e. A = (1 , 2 , 3 , 4 , 5 , 6). A natural game to play is to roll two of the dice, and declare the winner to be the one which has the highest number of spots showing on the top face. (If they are equal, then there is no winner.) A die is stronger than another if it wins with a higher probability. If two dice win with equal probabilities, we say they are tied. [a] Which of B and C are stronger? Or are they tied? [b] Which of C and D are stronger? Or are they tied? [c] Which of D and B are stronger? Or are they tied? [d] Why are the results of parts [a], [b], and [c] surprising? 5. Extra Credit (2 points) Let { A 1 ,A 2 ,A 3 ,... } be a sequence of events. ²ind an expression (set-theoretic) for the event that inFnitely many of the A i ’s occur. Justify your answer. 1...
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