Friends A and B are playing a card game. For any n=0, 1, 2, ..., let Esubn be the event in which A first won n-1 times in a row before B finally won a game. Afterwards, A claims that she keeps winning because she is better at the game. B claims that he is just as good as A, but A has been lucky. Suppose that A will win each game independently with probability p. If B is right, then p = 0.5, while is A is right then p=0.8.

a) Suppose n=1. What is the probability of Esubn is A is right? How about if B is right? (Yes, if they only played once and B won it doesn't make sense that they'd argue whether A is better, but let's just go with it)

b) Answer part (a) when n=2. How about n=3? Can you come up with a general formula for the solutions to part (a) given n?

c) Let X be the number of the first game that B wins, so if A wins 3 times in a row before B wins, then X=4. Suppose also that A is right. Using your answer to part (b), what is pmf of X, and what is the expected value of X?

d) X follows the geometric distribution. The geometric distribution is equipped with the parameter p and it measures the number of independent Bernoulli trials of probability p required to get a single success. In the above problem, what are the Bernoulli trials and what is a success? What is p if A is right? How about is B is right? (Note: if A is right, p is not 0.8).

e) Can you come up with another example of a random variable that is geometric (one that is unrelated to games between A and B)?

f) In R, the command rgeom(nap) returns a vector of length n of independent samples from a geometric distribution with parameter p. Use this and the information from the first two R tutorial to estimate the standard deviation of X when A is correct. Do the same for when B is correct. Use at least 10000 samples to do the calculations. Compare this to the theoretical standard deviation of root(1-p) all over p, where p is your solution to pard (d).