Question

There are 21 cards labeled with each of A, B, C,1,2, ..., 9. For each of A, B, C, there is one card labeled with it, and for each of 1, 2, ...,9, there are two cards labeled with it. Consider to repeat trials each of which is "to draw one from 21 cards uniformly at random.” Answer the following questions. (1) Let X denote the number of trials repeated until any one of A, B, C appears. Find the expectation of X. (2) Let Y denote the number of trials repeated until each of A, B, C appears at least once. Find the expectation of Y. (3) Let Z denote the number of trials repeated until each of 1, 2, ..., 9 appears at least once. Establish which is greater, the expectation of Z or the expectation of Y. (4) Let P denote the probability that each of A, B, C appears at least once after n trials. Describe P using n.

Answer 1

The answer is given below

x denotes the number аич - А.В. Фреон. of trials repeated untile (1 E(X) У , 5xPCX-7). X=0 го. Px = 0) + 1. Р(х - 1) + 2. Pix= 2) + * 1 (3) + 2. (18) 13,) + 2 ( 18 )*(3) + .. Е [+2. 6 + 3 (4) ] 2 x x4 - 1... (А). (2) y denote the no of trials until each of А, В, t арреаи ба . Let, У А+В +e. at A no of trial untill one of A, B, C occurs р. по 4 tried aн ух! 60 4 АВС occurs from the remaing offer first one ( wo of trials after second for the Ср Че 4 и А,Ве Есе) : E (4) t E (0) +€00 . е. Р(А: 1) +2 P (A - 2) t... . + 1. PCB = 1) + 2P (0:0) +- 1. PCe=1) + P ( = 2) + ... . /11/19 о

|- 1.(一) + 8 (15) 等) - s ()() | 41.() + 2 ())s (19) ); | tt (5) + 2 ()() + ( (幼) : 4.34 + 2017 : A+ + 發2), : 39. • 90 ...(A)。

3 . let a denotes the he no of trial until each of 2 9 . appears at least once 모(2) - 1.18 + R. 3 48. + 3.3 (13) | +1 + 2 (S) () + 3.15) (S +1 + 2 () ()+ 3 () ()+ t.S+ 2.())+ 3 (S) (1) + 1.(1) + 3 ()(1) t1. (0) + 2. (3)을) + 1 + 1. ( ) + 3 | ) () + 1 () + (1) () + t1. (S) + 2 (12)() | + it top + i = ) | : + + + + 월, ++ | 2 2 70 2 30.(Aw), :EY) is greater), 2020/1/19 0

- d) p denote the probability that each of A, Bye породил од Хоолл! one op -лч 7 4ъjals , Ut, & alo denotes the ADD no of that A, B, e appears in n trials :. the probility function is 2=0 =P(),1).