Geometric distribution

Two palyers battle in a competition and they each have 50% of winning 1 point at each round. The winner of the tournament is the first who is 
(a) two points ahead
(b) three points ahead
 Let X be the number of rounds played. Calcuate E[X], the distributuion of P(X=n), and Var[X] in the cases of (a) and (b)

Here is what I thought for part (a), but I am not quite sure about (b).

1st and 2nd rounds can be seen as a mini match. In the same way, 3rd and 4th is also a mini match. 5th and 6th is also a mini match and etc. The game ends when the first mini match is not a draw. There are two situations to end the game. 1. Player A wins. If player A wins, the probability of wining that last mini match is 1/4 (1/2 * 1/2). 2. Player A loses, it is the same probability (1/4). Therefore, the total probability of final mini match is 1/2 (1/4 + 1/4). Based on this pattern, the number of rounds X ~ 2Geo(1/2). E(X) = 4 Var(X) = 8.

  • Hi, I have a nice way to give you P(X=n) for case (b), but probability is not my field and to get E[X] and Var[X] I need more time than what the offered amount is worth. Is it okay if I just give you P(X=n)? It seems you know by heart how to get E[X] and Var[X] yourself. Otherwise I can do that if you increase the offered amount.

  • Honestly also just finding P(X=n) would be worth more than the value of the question, but I thought it was an interesting one to answer and so I computed that regardless. Let me know your preference!

  • I can answer this question, but your offered amount is too low.

  • What level is this? Are we allowed to use Markov Chains?

  • There is no need, I have a very elementary solution, I just don't want to spend time computing expected value and variance for this little money.

  • Yes, you can use Markov Chain. It is my first time using this website, I did not know how much is low amount and how much is high amount. I I put most of the money on another question (card shuffling). If you want to get more, you can have a look at that one. I have worked out part a in this question (2 points ahead), so I just need (b). To be more specific, I just need the distribution for b. Thank you very much for your help!

  • I have the distribution for b, if you're ok with just that I will answer.

  • But please confirm!

  • Sure.you can tell me the distribution. I just wrote my answer my part b. X ~ 2Geo(1/4) +1. E(X)= 9, Var(X)= 48. I guess this makes your situation a bit easier.

  • So I can see if I am correct or not.

  • I get a different answer. I'll post it.

  • I don't think this make sense to me. In the three points situation, we can regard 2nd&3rd, 4th&5th, 6th&7th ..... rounds as mini-match, which means that 1st round is seprerated and not being included in the mini match. Therefore, there are two situations happening in the game: 1. the points difference is either 1point. 2. the game ends becasue the person who led the 1 point wins the final two mini match (p=1/4). Therefore, all the mini matches follows Geo(1/4), so X~2Geo(1/4)+1.

  • that plus 1 means we plus the first round that did not get included in the mini matches.

  • I think we're saying the same thing, except that I don't know the notation for the distribution (which I already admitted). The probability of the match ending after 2n+1 rounds is still (3/4)^(n-1) * 1/4, which you counted as "not finishing in the first n-1 mini-matches times finishing in the n-th", which is the same thing I said (but counted in a different way).

  • I checked the definition, and it is indeed 2 Geo(1/4) + 1, which gives P(X=2n+1) = (3/4)^(n-1) * 1/4. My bad for not checking before, but the answer I gave is also correct.

  • Ok, cool! Thank you very much for confirming this!

  • No worries! Also your solution is more elegant than mine, I have to say!

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