forum.virtualchallengemeets.com

Alex, Mel, and Chelsea play a game that has [unparseable or potentially dangerous latex formula] rounds. In each round there is a single winner, and the outcomes of the rounds are independent. For each round the probability that Alex wins is [unparseable or potentially dangerous latex formula], and Mel is twice as likely to win as Chelsea. What is the probability that Alex wins three rounds, Mel wins two rounds, and Chelsea wins one round?

nsguy can you post the answer too so I don't look dumb while answering it. OK this is how I would do it(Just a warning, its probably completely wrong)

_ _ _ _ _ _ There are Six Rounds.
A has a 1/2 chance to win a round
M has a 1/3 chance to win a round
C has a 1/6 chance to win a round
1/6+1/3 = 1/2
1/3 = 2(1/6).


The Number of 3 round combinations is 6 C 3 = 20.
The Number of 2 round combinations is 3 C 2 = 3. You have to account for the three rounds A has already won so its 3 choose 2 not 6 choose 2.
The Number of 1 round combinations is 1 C 1 = 1. There is only one more round to win.


Now I just calculated the probability of winning three straight for A to be 1/8. I multiplied this by the number of 3 round combinations to get 5/2. The probability for winning 2 straight for M is 1/9. (1/9)(3) = 1/3. Finally there is one game to be won. C has a 1/6 chance of winning it.
I multiplied the results together to get:
(5/2)*(1/3)*(1/6) =



Its probably completely wrong, but it seemed reasonable to me. This problem wasn't that time consuming. Sadly its one of the few problems I can probably solve on an AMC test. AMC seems really fun though. I love problems that make you think and don't just make you apply a formula.

I was going to post all of the problems, then answer some later. I only posted 11 because I wanted to go do something else, but I'll post more later (and I skipped 2 since it had a diagram that I could not describe without giving away the solution). I decided AMC 12 might be better to post than AMC 12.

I'm copy-pasting the problems from http://www.theartofproblemsolving.com. I do my best to not look at the solution (the problem and solution are on the same page) and copy/paste the problem on here, inserting LaTeX where necessary. If you want to check your answers, I suggest going there. Solutions to the problems are included on there, but I suggest at least attempting the problem first, and don't post the solution if you didn't come up with it, please.

Also, use LaTeX as it makes it a bit easier to read. Instead of bolding your answer in large font, boxing in LaTeX is better; [unparseable or potentially dangerous latex formula] works.

Okay. I was unsure if you wanted it answered or not, but I went ahead since I was kinda bored. Plus, I wanted to know how would I do, solving these problems differently then I normally would. I was surprised I got the first 10, right. I got the 11th wrong though, since probability is definitively my weak point....

But I guess I should do a practice test now to see how would I do. I've never done this before, so I'm going into it blindfolded, haha.