- Alice Is Bored And Decided To Play The Following Game She Has A Regular Deck Of 52 Different Cards That She Placed On H 1 (110.8 KiB) Viewed 39 times
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Alice is bored and decided to play the following game. She has a regular deck of 52 different cards that she placed on her desk face up and next to each other. She also has a program that simulates drawing a card at random from a similar deck, more precisely, each time she uses the program it shows one of the 52 possible cards with probability 1/52. Don't worry, Rupert always goes back home when it's time for supper, he won't lose a meal! However, you can't use this information to solve the question. 3 a Alice then proceeds to virtually draw the first card in the program, and she puts a chip on the corresponding card on her desk. From the second draw onward, she virtually draws a card in the program and looks at her desk. If there is no chip over the corresponding card on her desk, she places a chip on it; otherwise she just draws a new card. Alice continues playing until she places a chip on the last card on her desk. Consider the following example in which the first 5 rounds of a game are de- scribed. 14 Alice draws the 1card in her program and it is a 2. Alice places a chip on the in her desk; 2nd. Alice draws the 2nd card in her program and it is a Alice places a chip on the in her desk; 3rd. Alice draws the 3rd card in her program and it is a Alice places a chip on the in her desk; 4th: Alice draws the 4th card in her program and it is a Alice notices that there is already a chip on the in her table, thus no chip is placed in this round; 5th. Alice draws the 5th card in her program and it is a ), Alice places a chip on the on her desk. Let X denote the random variable that counts the number of cards Alice has virtually drawn with her program. Prove that E(X) satisfies the following in- equalities: n ln(n) < E(X) n(In(n) +1), with n = 52.