AMC 12A 2012 #11

[nsguy1350]

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?

[sxk1693]

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.

[nsguy1350]

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.

[darksaber21]

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.