A mp3 player contains 10 different songs in its memory and chooses songs randomly (every choice is independent on the previous choices).Let X be the number of songs one will hear until the first time (including this time) the same song will be chosen twice in a row .A. Write the probability function of XB. What is the expectation of X?

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Alexia Mata · Accepted Answer

Step 1The number of ways to list 10 different songs k times, where in the first   k  −  2 songs no 2 consecutive songs are identical and the last 2 songs are identical, is equal to the product of:- 10 options for song #1- 9 options for songs   #  2  ,  …  ,  #  k  −  1- 1 option for songs #kStep 2The total number of ways to list 10 different songs k times is       10    k  .Hence the probability function is:  P  (  X  =  k  )  =            10      ⋅              9                  k          −          2                    ⋅      1              10      k        =            9              k        −        2                    10              k        −        1

odigavz · Accepted Answer

Step 1The random variable   Y  =  X  −  1 will be geometric with parameter 0.1. (You can&#039;t get a match on the first song; but after that, the probability of a match with the immediately previous song is 0.1 on each trial.)Step 2So   P  (  X  =  x  )  =      0.9          x      −      2        ⋅  0.1 for   X  =  2  ,  3  ,  4  ,  …And   E  [  X  ]  =  1  +  E  [  Y  ]  =  1  +      1    0.1    =  1  +  10  =  11

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