In the Harry Potter world, each Chocolate Frog has a card inside it for one to...

  

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In the Harry Potter world, each Chocolate Frog has a card inside it for one to collect famous witches and wizards (a.k.a., Chocolate Frog Card). In Harry's first journey on the Hogwarts Express, he got his very first Chocolate Frog Card, which is Albus Dumbledore. Assume that there are N possible wizards or witches to appear in the Chocolate Frog Cards and each unique card exhibits a equal chance (A) to be drawn in a Chocolate Frog. (a) [Non-coding Question] Harry collected another X cards till he got another Dumbledore (i.e., Dumbledore is the X-th card while none of the previous X - 1 cards is a Dumbledore). If there are 3 possible unique Chocolate Frog cards (N = 3), what is the expected value of X (i.e., E[X])? (10') (b) [Non-coding Question] If we didn't set N to an exact number, please generalize the solution of (a) to represent E[X] using a formula with N. (10') (c) [Coding Question] On the Hogwarts Express, Ron Weasley, one of Harry's best friends, told Harry that he collected 500 cards while he has not yet got Agrippa or Ptolemy. Assume that N = 100, write the R codes to run a Monte Carlo Simulation to estimate the probability that one has 500 cards (with possible replications) while neither Agrippa nor Ptolemy is included. (10') In the Harry Potter world, each Chocolate Frog has a card inside it for one to collect famous witches and wizards (a.k.a., Chocolate Frog Card). In Harry's first journey on the Hogwarts Express, he got his very first Chocolate Frog Card, which is Albus Dumbledore. Assume that there are N possible wizards or witches to appear in the Chocolate Frog Cards and each unique card exhibits a equal chance (A) to be drawn in a Chocolate Frog. (a) [Non-coding Question] Harry collected another X cards till he got another Dumbledore (i.e., Dumbledore is the X-th card while none of the previous X - 1 cards is a Dumbledore). If there are 3 possible unique Chocolate Frog cards (N = 3), what is the expected value of X (i.e., E[X])? (10') (b) [Non-coding Question] If we didn't set N to an exact number, please generalize the solution of (a) to represent E[X] using a formula with N. (10') (c) [Coding Question] On the Hogwarts Express, Ron Weasley, one of Harry's best friends, told Harry that he collected 500 cards while he has not yet got Agrippa or Ptolemy. Assume that N = 100, write the R codes to run a Monte Carlo Simulation to estimate the probability that one has 500 cards (with possible replications) while neither Agrippa nor Ptolemy is included. (10')