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stat2303_tutorial01

stat2303_tutorial01 - THE UNIVERSITY OF HONG KONG...

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Page 1 of 5 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCES STAT2303 Probability Modelling (2010-2011) STAT2803 Stochastic Models (2010-2011) Tutorial 1 Solutions 1. A and B roll a pair of dice in turn, with A rolling first. A’s objective is to obtain a sum of 5 while B’s objective is to obtain a sum of 9. The game ends when either player reaches his objective, and the player is declared the winner. (a) Find the probability that A is the winner. P(Sum of a pair of dice = 5) = 4 36 = 1 9 P(Sum of a pair of dice = 9) = 4 36 = 1 9 P(A wins the game) = P(A wins in his 1st roll) + P(A wins in his 2nd roll) + P(A wins in his 3rd roll) + = 1 9 + 8 9 × 8 9 × 1 9 + 8 9 × 8 9 × 8 9 × 8 9 × 1 9 + = � � 8 9 2n 1 9 n=0 = 1 9 1 1 (8 9 ) 2 = 9 17 (b) Find the expected number of rolls of dice until A or B wins. Let 𝑁𝑁 be the number of rolls required until A or B wins. Method 1 𝐸𝐸 ( 𝑁𝑁 ) = � 𝑛𝑛𝑛𝑛 ( 𝑁𝑁 = 𝑛𝑛 ) 𝑛𝑛 =1 = � 𝑛𝑛 ( 𝑁𝑁 > 𝑛𝑛 ) 𝑛𝑛 =0 = 𝑛𝑛 ( 𝑁𝑁 > 0) + 𝑛𝑛 ( 𝑁𝑁 > 1) + 𝑛𝑛 ( 𝑁𝑁 > 2) + = 1 + [1 − 𝑛𝑛 ( 𝑁𝑁 = 0) − 𝑛𝑛 ( 𝑁𝑁 = 1)] + +[1 − 𝑛𝑛 ( 𝑁𝑁 = 0) − 𝑛𝑛 ( 𝑁𝑁 = 1) − 𝑛𝑛 ( 𝑁𝑁 = 2)] + = 1 + 1 0 1 9 + 1 0 1 9 − � 8 9 � � 1 9 �� + 1 0 1 9 − � 8 9 � � 1 9 � − � 8 9 2 1 9 �� + = 1 + 8 9 + 64 81 + 512 729 +
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Page 2 of 5 = 1 1 8 9 = 9 Method 2 𝐸𝐸 ( 𝑁𝑁 ) = � 𝑛𝑛𝑛𝑛 ( 𝑁𝑁 = 𝑛𝑛 ) 𝑛𝑛 =1 = � 𝑛𝑛 � 8 9 𝑛𝑛− 1 1 9 𝑛𝑛 =1 = ( 𝑛𝑛 + 1) 8 9 𝑛𝑛 1 9 𝑛𝑛 =0 8 9 𝐸𝐸 ( 𝑁𝑁 ) = � 𝑛𝑛 � 8 9 𝑛𝑛 1 9 𝑛𝑛 =1 𝐸𝐸 ( 𝑁𝑁 ) 8 9
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