Probability Question

Question

I've got a probability question from my Princeton Math Workout book that I don't quite get the logic of. It reads:

Three children, John, Paul, and Ringo, are playing a game. Each child will choose either the number 1 or the number 2. When one child chooses a number different from those of the other two children, he is declared the winner. If all of the children choose the same number, the process repeats until one child is declared the winner. If Ringo always chooses 2 and the other children select the numbers randomly, what is the probability that Ringo is declared the winner?

I came up with 1/4. Since Ringo is always going to pick 2, it seems like we have the following possibilities:

John: 1
Paul: 1 (Ringo wins)

John: 1
Paul: 2 (John wins)

John: 2
Paul: 2 (no winner)

John: 2
Paul: 1 (Paul winner)

So there are four possible outcomes, one of which involves Ringo winning.

The books says the answer is 1/3, though. It seems to be excluding the possibility of there being no winner at all. Here's its explanation:

So, I don't know which logic to follow. I understand what the book is saying--that if there's a winner, the odds are 1 out of 3 that it will be Ringo. But there's no guarantee that there will be a winner at all; there's a 1 out of 4 chance that there won't be. And that would change Ringo's odds of winning from 1/3 to 1/4.

Is this sort of confusion typical of a real GMAT question?

abhijit_sen · Answer

Question is asking when finally Ringo do Win. So for Winning (anyone of them wins) there is only 3 possible ways and nothing else.

ParrotHead · Answer

But the question asks, "If Ringo always chooses 2 and the other children select the numbers randomly, what is the probability that Ringo is declared the winner?"

To me, that's not the same as asking, "If Ringo always chooses 2 and the other children select the numbers randomly, what is the probability that the winner will be Ringo?"

The answer to the first question would be 1/4. The answer to the second question would be 1/3.

I may not be good at math, but I  am  pretty good at English (it was my major). I'm hoping that the real GMAT won't have any questions phrased so confusingly!

shrenik222shah · Answer

Hi,

Here we have to omit the case of all selecting the same number ie all select 2 as in that case the round will be repeated.So we have threee possibilities as posted by the first poster and probablity turns out to be 1/3 and not 1/4.

ParrotHead · Answer

Clearly, Ringo has a 1/3 chance of being the winner--assuming there is a winner. But I don't like to assume things. Based on how the question was worded, how would I know to make that assumption? Because the reality is that in any round of the game, there is a 1/4 chance of there being no winner at all.

I think I'm just frustrated because everything I've learned so far in my studies for GMAT has taught me never to make assumptions when answering questions. If a stem doesn't explicitely state something, I shouldn't assume it to be so.

With this item, the stem doesn't explicitely state that we're looking for the odds of Ringo being the winner in a round where there is a 100% chance that someone will win. Therefore, there's no reason to think that we're not talking about a normal round--one in which there are 4 possible outcomes, 1 of which is that no one wins.

Now, it could be that this stem is just weak and unique to the Princeton book, and wouldn't actually appear on the GMAT worded this way. I sure hope that's the case.

ngocanh88 · Answer

I am a bit confused about the wording. Can we understand like: Ringo wins when the numbers are 211. So, its probability is 1*1/2*1/2=1/4
Thanks

fivedaysleft · Answer

aaah...this is a problem involving bayes theorem of probability.
event of ringo winning is two events exclusive of each other = there is a winner X ringo wins 
similiarly paul wins = winner X paul
 john wins = winner X john

winner = 3/4 = .75 
john = paul = ringo = 1/4 = .25

now, the probability of ringo winning is = event when ringo wins / (sum of all winners)
 = .75*.25 / ((.75*.25)+(.75*.25)+(.75*.25))
 =1/3 (clearly! no calculations needed)

the catchphrase here in the question is "If all of the children choose the same number, the process repeats until one child is declared the winner"

fivedaysleft · Answer

nice question! +1

hope my explanation helped.

ngocanh88 · Answer

Got you, a good one, kudos

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