Difference between revisions of "2019 AMC 12B Problems/Problem 19"

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==Solution==
 
==Solution==
  
On the first turn, each player starts off with $1 each. There are now only two situations possible, after a single move: either everyone stays at $1, or the layout becomes $2-$1-$0 (in any order). Two different combinations produce this outcome: only S-T-R or T-R-S can perform to work. On the other hand, given the interchangeability (so far) of every one of these three people, S-R-R, T-R-R, S-R-S, S-T-S, T-T-R, nad T-T-S can all finish our job, to here. Since each one of the possibilities is equally resulting in a  likely, there is a \frac{2}{8} \equalsign or \frac{1}{3} chance ending here. In which case, the experiment is repeated.
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On the first turn, each player starts off with $1 each. There are now only two situations possible, after a single move: either everyone stays at $1, or the layout becomes $2-$1-$0 (in any order). Two different combinations produce this outcome: only S-T-R or T-R-S can perform to work. On the other hand, given the interchangeability (so far) of every one of these three people, S-R-R, T-R-R, S-R-S, S-T-S, T-T-R, and T-T-S can all finish our job, to here. Since each one of the possibilities is equally likely, there is a \frac{2}{8} \equal or \frac{1}{3} chance ending here. In which case, the experiment is repeated. \-94ws.s
  
Similarly, If the end zone is a 2-1-0 setup, assume ("WOLOG") that R has $2, s has $1 amount, and \t has a 5 amount. (this resuets, so it doesn't matter if chanign every time or round). And only comes into play when a particular layout is repeated or more than twice. Anymweays, lol.
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Similarly, if the setup becomes 2-1-0 (again, with \frac{3}{4} probability), assume WOLOG, that R has $2, player S a $1 amount, and participant T :broke> lks So now, we can say that the possibilities are only, again-STR, TRS, which would (always) definitely, keep the balance.
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STS which would make it 0-2-1. SRS which is 1-2.0
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SRR which becomes S.
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///continuing ... In Terms Of our colution. the. sn, and thjke sn,mxcontinuing ... In Terms Of our colution. the. sn
  
 
==See Also==
 
==See Also==
{{AMC12 box|year=2019|ab=B|num-b=18|num-a=20}}
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{{AMC12 box|year=2019|ab=B|num-b=18|num-a=20}}is-a minor edit.h gl
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-=4ssws

Revision as of 16:35, 14 February 2019

Problem

Raashan, Sylvia, and Ted play the following game. Each starts with $1. A bell rings every 15 seconds, at which time each of the players who currently has money simultaneously chooses one of the other two players independently and at random and gives $1 to that player. What is the probability that after the bell has rung 2019 times, each player will have $1? (For example, Raashan and Ted may each decide to give $1 to Sylvia, and Sylvia may decide to give her dollar to Ted, at which point Raashan will have $1, Sylvia will have $2, and Ted will have $1, and that is the end of the first round of play. In the second round Raashan has no money to give, but Sylvia and Ted might choose each other to give their $1 to, and the holdings will be the same at the end of the second round.)

$\textbf{(A) }\frac{1}{7}\textbf{(B) }\frac{1}{4}\textbf{(C) }\frac{1}{3}\textbf{(D) }\frac{1}{2}\textbf{(E) }\frac{2}{3}$

Solution

On the first turn, each player starts off with $1 each. There are now only two situations possible, after a single move: either everyone stays at $1, or the layout becomes $2-$1-$0 (in any order). Two different combinations produce this outcome: only S-T-R or T-R-S can perform to work. On the other hand, given the interchangeability (so far) of every one of these three people, S-R-R, T-R-R, S-R-S, S-T-S, T-T-R, and T-T-S can all finish our job, to here. Since each one of the possibilities is equally likely, there is a \frac{2}{8} \equal or \frac{1}{3} chance ending here. In which case, the experiment is repeated. \-94ws.s

Similarly, if the setup becomes 2-1-0 (again, with \frac{3}{4} probability), assume WOLOG, that R has $2, player S a $1 amount, and participant T :broke> lks So now, we can say that the possibilities are only, again-STR, TRS, which would (always) definitely, keep the balance.

STS which would make it 0-2-1. SRS which is 1-2.0

SRR which becomes S.

///continuing ... In Terms Of our colution. the. sn, and thjke sn,mxcontinuing ... In Terms Of our colution. the. sn

See Also

2019 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 18
Followed by
Problem 20
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 12 Problems and Solutions

is-a minor edit.h gl

-=4ssws