06_02 - n n1,n m = n n1!n m p 6-11 ways Example MISSISSIPPI...

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p. 6-11 n n 1 , ··· ,n m = n ! n 1 ! ×···× n m ! ways . Example: MISSISSIPPI 11 4 , 1 , 2 , 4 = 11! 4!1!2!4! . Example (Die Rolling). Q : If a balanced (6-sided) die is rolled 12 times, P (each face appears twice)=?? Sample space of rolling the die once (basic experiment): 0 = {1, 2, 3, 4, 5, 6}. The sample space for the 12 trials is = 0 × × 0 = 0 12 An outcome is =( i 1 , i 2 , …, i 12 ), where 1 i 1 , …, i 12 6. There are 6 12 possible outcomes in , i.e., # = 6 12 . Among all possible outcomes, there are of which each face appears twice. P (each face appears twice) = 12 2 , 2 , 2 , 2 , 2 , 2 = 12! (2!) 6 12! (2!) 6 1 6 12 . p. 6-12 Generalization. Consider a basic experiment which can result in one of m types of outcomes. Denote its sample space as 0 = {1, 2, …, m }. Let p i = P (outcome i appears), then, (i) p 1 , …, p m 0, and (ii) p 1 + + p m = 1. Repeat the basic experiment n times. Then, the sample space for the n trials is = 0 × × 0 = 0 n Let X i = # of trials with outcome i , i =1, …, m , Then, (i) X 1 , …, X m : R , and (ii) X 1 + + X m = n . The joint pmf of X 1 , …, X m is p X ( x 1 ,...,x m )= P ( X 1 = x 1 ,...,X m = x m ) = n x 1 , ··· ,x m p x 1 1 ×···× p x m m . for x 1 , …, x m 0 and x 1 + + x m = n .
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p. 6-13 Proof. The probability of any sequence with x i i ’s is and there are such sequences.
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06_02 - n n1,n m = n n1!n m p 6-11 ways Example MISSISSIPPI...

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