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12 pfive 6 and five odd pfive even and five odd 10 7

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12 P(five "6" and five odd ) P(five even and five odd) 10! 7 5 5! 5! 12 12               
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Q. 1 (20 pts) (MT1-2007) Assume two four-sided biased dice are rolled (each die has 4 faces) and their absolute difference is used to determine the number of tosses to make by a biased coin ( Head or Tail ) in the next step (if the difference is 0, then a single toss is made). In every trial, it is strictly necessary to stop the experiment, if the result of the current toss is equal to that of the last toss, even if the required number of tosses is not reached. Assuming that the ordered set of tosses is observed as an outcome (e.g. “… HT …”), a) Write the sample space for this experiment. b) Assuming that two subsets of the sample space are defined as E 1 and E 2 , write the smallest field, , which contains these two sets. c) Let E 1 = (“ number of heads number of tails ”) and E 2 = (“ number of heads 1”). Find the probability of set A = { H , TH , HT }, if the following event probabilities are given P( E 1 ) = 0.62 and P( E 2 ) = 0.73 (Hint . Represent set A in terms of sets E 1 and E 2 .). a)   , , , , , , , , , T H TT HH TH HT THH HTT THT HTH  b)   1 1 2 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 , , , , , , , , , , , , , E E E E E E E E E E E E E E E E E E E E    c)   1 , , , , , E H HH TH HT THH HTH   2 , , , , , , E T H TT TH HT HTT THT 12 A E E          1 2 1 2 1 2 1 A P E E P P E P E P E E              1 0.62 0.73 1 0.35 P A P E P E    
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1.( 20 pts) A fair thetrahedral die is rolled and a ball is drawn from one of the four boxes according to the result of die rolling. It is given that each box contains different number of red and white and blue colored balls as follows: Box-1: 4 red, 4 white, no blue Box-2: 2 red, 6 white, 4 blue Box-3: no red, 5 white, 10 blue Box-4: 2 red, no white, 6 blue a) What is the probability of drawing a white colored ball? 3 1 3 1 2 1 2 1 4 1 ) ( ) | ( ) ( 4 1 = + + = = = i P i w P w P i b) What is the probability of drawing a white colored ball given that die rolling resulted in an odd number? 2 1 }) 3 , 1 ({ ) ( = = P odd P , 24 5 4 1 3 1 4 2 2 1 ) 3 ( ) 3 | ( ) 1 ( ) 1 | ( }) 3 { } 1 ({ ( )) (( = + = + = = P w P P w P w P odd W P 12 5 2 / 1 24 / 5 ) ( ) | ( = = = odd odd w P odd w P c) If the selected ball is white, what is the probability that what is the probability that die rolling resulted in an odd number? 8 5 3 / 1 24 / 5 ) ( ) ( ) | ( = = = w P w odd P w odd P d) Given that the selected ball is white what is the probability that die rolling resulted in 3? 4 1 ) ( ) 3 ( ) 3 | ( ) | 3 ( 3 1 4 1 3 1 = = = w P P w P w P e) Given that the selected ball is white and it is returned back to the box it is drawn from, then a second ball is selected from the same box. What is the probability that the second selected ball is white too?
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