During each turn of a game, a player rolls fair six-sided die with

Question

During each turn of a game, a player rolls fair six-sided die with sides numbered 1 through 6 and adds the result to his or her cumulative score in the game. Then, next player takes a turn. The first player to get a cumulative score of 20 points or more wins. Torrance and Adia are playing the game. There are no other players. Torrance has 15 points and Adia has 17. It is Torrance's turn and he is about to roll the dice.

Let X equal the probability that Torrance will win the game during his current turn. Select for  First  and for  Second  the two operations such that applying the first to x and then applying the second to the result of the first would yield the probability that Adia will win on her next turn. Make only two selections, one in each column.

Bunuel · Accepted Answer

Torrance will win if he rolls a 5 or 6. Hence, the probability that Torrance wins is 2/6 = 1/3. Adia will win if Torrance fails to win, meaning if he does not roll a 5 or 6, and then Adia rolls a 3 or higher. Hence, the probability that Adia wins is 2/3 * 4/6 = 2/3 * 2/3. Now, the question asks to apply the First and Second operations to x, which is 1/3, such that we get 2/3 * 2/3. After some analysis, we can see that if we subtract x from 1, we get 1 - 1/3 = 2/3, and then multiply the result by 2/3, we get exactly what we wanted: 2/3 * 2/3. Hope it's clear. GMAT-Club-Forum-3sf1kerk.png

bb · Answer

So the fact that this is a probability question, already puts it into the top 20% of the hardest questions on the GMAT.

The second aspect is just the amount of reading you have to do and then wrapping your mind around what the question is actually asking. I had to read the second part a few times to understand it. I don't feel this is a typical question. I would say this is in the top 2--3% of the GMAT Questions by difficulty because of these factors and as the result, I would say it likely will not be easy to find a methodology to approach this question quickly. As you said, you can calculate probability faster by practicing but when you encounter a question you never faced before, it will ALWAYS take more time, which means you need to gain time on other easier ones so you have extra time needed for these harder ones. This is also how hard CR's work by the way - there are only a few patterns and it is because they do not have a pattern that they are hard, and they require extra time as the result as well.

abcsayali · Answer

Bunuel  can you explain how to solve this question?

To win Torrance need 5,6 on dice so probability x = 2/6 = 1/3
For Adia to win she needs 3,4,5,6 so probability = 4/6 = 2/3

I am not able to solve beyond this

Bismuth83 · Answer

You got probability x correct, however, for Adia, her probability to win is when she not only rolled 3,4,5,6 but also Torrance gets less than 20 after his roll.

In particular, for Torrance to get less than 20, he needs to get 1,2,3,4 - which has a  \frac{2}{3}  probability. Adia then needs to get 3,4,5,6, which as you said, has a probability of  \frac{2}{3} . That means the probability for Adia to win is rather  \frac{2}{3} \cdot \frac{2}{3} .

boybread5 · Answer

What I'm struggling with is what is the most efficient way to actually solve the problem. Figuring out that Torrance's probability is 1/3 and that Adia's probability is 4/9 is easy enough to figure out (probably with reading above and calculations, I would say takes 45 - 55 seconds). The issue then is what is the most efficient way to apply the probabilities and in ~30 seconds - 1 minute solve First and Second.

I initially went about solving the problem by going line by line, first with Add 1/2. The issue is this approach takes forever and ended up adding like 2+ minutes on just solving this piece.

Would be great to hear a systematic way others approach the last piece of applying these probabilities to actually solving the problem.

Dbrunik · Answer

I too struggled with this question but i think there is a more systematic approach here in addition to just fully testing out each of the cases, which as bb pointed out, is inevitable at a problem of this difficulty level. The approach lies in  fully & quickly understanding  how the given operations effect a fraction/probability (or really any operation). So using knowledge of number properties, fractions, and probability, here are some logical steps you can use to eliminate answers - since you know that you cant simply use algebra to target the correct answer you are going to have to rule some out!!

First, we know that we have 1/3, and we will apply two operations to this to arrive at 4/9. express this 4/9 in  both  4/9 and 2/3*2/3 on your paper, for clarity.

(its probably worth noting that if you weren't able to get to this setup, this question is going to be too difficult for you, and its best you begin with easier probability questions)

given this, we know that our TWO operations have to have a positive effect of  + 1/9  in order for them to equal. obviously, any operation that makes our fraction smaller will not be correct. (anything resulting in two (-)(-) operations can be eliminated)

start by understanding the operations 
 add 1/2 : obviously this increases our fraction
 divide by 3:  dividing a fraction makes it smaller
 Multiply by 2/3 : multiplying a fraction by another fraction will always make the resulting fraction smaller. 
 subtract 1/6 : obviously this makes our fraction smaller. 
 Subract from 1 : hmmmm this one is interesting when compared to the others because depending on if you've determined the initial probabilities correctly, this operation will give you a value their can either increase or decrease.  This tells me this answer choice will probably be necessary, since it adds a layer of complexity that the others do not . if you subtract any fraction from 1, it is effectively the inverse of the fraction. 100/100 - 33/100 = 66/100 or 100/100-66/100= 33/100 since we have a fraction LESS THAN 50%, subtracting it from 1 will result in a LARGER VALUE. (Subtracting a fraction GREATER than 50% from 1 will result in a SMALLER fraction)

Now, just given these, you could quickly go through and eliminate possible answer combinations that ONLY REDUCE the fraction. 
examples: divide by 3 and multiply by 2/3 are out, multiply by 2/3 and divide by 3 are out. divide by 3 and subtract 1/6, multiply by 2/3 and subtract 1/6.

additional examples that are easy to eliminate are adding 1/2 and subtracting 1/6, in either order. you should be able to do these in your head relatively quickly. and deduce that neither arrives at 4/9.

also, think again quick about subtracting 1/6 then subtracting from 1. your fraction will obviously be far too large. - eliminate.

given that none of these work, i know i'm going to be starting with A or E

lets start with E first (for "first"), since its so unique from the others. 
if we subtract from 1 and add 1/2, we will be too large - eliminate
if we subtract from 1 and divide by 3, our fraction will again be too small (2/3*1/3 = 2/9) this should be pretty intuitive - eliminate 
if we subtract from 1 again we get 2/3, hmmm now multiply by 2/3?  bingo , we have our 2/3*2/3. This is why its important to leave the simplification as 2/3*2/3 and not to just write 4/9. sometimes its better to see things in all their forms instead of in the most simplified.

#
This means that any operations that have a negative effect can be eliminated. start from t
#

Dbrunik · Answer

check out my other post above that aids a little bit in how to approach these complex TPA probability questions.

KarishmaB · Answer

It is a question in which the 2nd answer depends on the 1st. Invariably, such questions are a little harder.

Getting the probability of their wins is the easy part here.

Torrance has 15 points and needs a 5/6 to get 20 or more. Hence his probability of winning = 1/3
Adia has 17 points. She wins if Torrance loses on this turn and Adia wins on hers. Adia needs 3/4/5/6 to win on her turn. Probability = 2/3 * 4/6 = 4/9

What two steps do we need to convert 1/3 to 4/9? Of course, it can be done in infinite ways. So let's focus on the available options.

Add 1/2

Not the step 1. It will give us 2 as a factor in the denominator which we do not need.

Divide by 3

1/3 becomes 1/9 but then we need to add 3/9 to get 4/9. Adding 3/9 is not an option. So this cannot be step 1.

Multiply by 2/3

If this were the first step, we would get 2/9 so we would need to further multiply by 2. That is not an option.

Subtract 1/6

Again, brings 2 as a factor in the denominator. Problem.

Subtract from 1

When we subtract 1/3 from 1, we get 2/3. This looks promising. We have "Multiply by 2/3" in the option (C) which can be the second step.

Select 
 First:  Subtract from 1
 Second:  Multiply by 2/3

Note: In the more difficult questions, the first part is often lower placed than the second. I like to start from the bottom in these questions though it is mostly a roll of die!

johnobrien1321 · Answer

I just want to complain that the question is confusingly worded. "... the probability that Adia will win on her next turn .." is ambiguous; it could be interpreted as "... the probability that Adia will win on her next turn  ..." or could be interpreted as "... the probability that Adia will win on her next turn  ". It is the confusing way they phrase the question IMHO that makes is harder than it has to be.

Nipunh · Answer

this is how I did it under 2.5 mins,
Step 1) Getting the value of X = 1/3 and the final value as 4/9 was a pretty straight forward step, that took me around 40 seconds to write down.
Step 2) Eyeballing the data and trying to make some sense out of it... I have 1/3 and with 2 operations I am supposed to reach 4/9... cool ,p.s. this is how I denoted this ((1/3){}) {}=4/9
Step 3) Thinking how it could be easier / planning on how to solve based on the data. Thought of solving it backward, as I saw multiply by 2/3 ... (also adding and subtracting seemed like a tough journey ahead), instantly I thought this could be a good starting point from back, if I have to brute force my answer I would like to bet on this.
Step 4) checked whether my bet payed off or not by checking options wrt this choice... Viola! I got a hit. If I wouldnt have got a hit, I would have seen any other obvious suspects, if not, I would have left this in a test scenario.
Step 5) Mark and move on

Goldengrams · Answer

johnobrien1321

I agree that this question is worded incorrectly. The question asks for the probability that Adia will win "on her next turn" which implies she actually takes a turn. It does not say “if Torrance fails” or suggest any conditional dependency. Therefore, the natural interpretation is that Adia’s turn is guaranteed, and her win probability is simply the chance she rolls 3 or higher. Using (1−x)⋅(2/3) adds an unstated assumption and answers a different question than the one actually asked.

On the GMAT, we are trained to avoid assumptions, follow wording carefully, and not infer more than is stated. This question punishes you for doing exactly that.

cheshire · Answer

I understand the concern about adding an unstated assumption, but the condition “if Torrance fails to reach 20” is not an extra assumption and is built into the very phrase “on her next turn” in a turn-based game. In any turn-by-turn context: A player only gets a next turn if the game hasn’t already ended.

So when it asks for Adia’s probability of winning on her next turn, it means:
 
 Torrance takes his turn first. 
 If Torrance reaches 20 or more, the game ends immediately (and Adia never rolls). 
 Only if Torrance does not win does Adia get to roll, and then her chance of winning is 4/6=2/3

Goldengrams · Answer

@chesire

How do we know the question isn’t instead asking for the probability if it is already Adia’s turn? In that case, Torrance has already failed, and the question is “What’s her probability of winning now?”

Both interpretations are logically valid, though one is a literal interpretation of the text and the other requires an assumption added on top.

cheshire · Answer

If Adia’s turn were already underway, there would be no need to introduce X at all (since X only pertains to Torrance). Likewise, “on her next turn” in a strictly turn‐based game means “on the turn that follows the one we’ve just been talking about” and here that “just been talked about” turn is Torrance’s.

Goldengrams · Answer

Introducing x could just as easily serve to define a reference point for applying operations, not to imply a causal link.

We’re explicitly told to apply operations to x, not that Adia’s outcome is conditioned on it. But “on her next turn” without any condition reads, in plain language, as a guaranteed event. I’m not saying your interpretation is illogical, but it’s not a necessity dictated by the language. It adds a dependency that isn’t actually stated, which makes the answer ambiguous.

cheshire · Answer

"On her next turn" is included because it is possible that she could win two turns from Torrance's turn, which is outside the scope of the problem.

The key information here is that it is Torrance's turn and we are calculating the probability that Adia wins the game by her next turn.

Considering Torrance is about to roll the dice, it makes no sense to calculate a probability that doesn't include all the variables and only includes Adia.

I see what your trying to argue here, and while It’s true that in everyday speech “on her next turn” might sound like a guaranteed event, in the context of  GMAT's  sequential, turn‐based probability questions, its not. The “next turn” is inherently contingent on the game not having already ended as we are looking at the game as whole, not just Adia's turn in isolation.

navs18 · Answer

There is a gap here, it is no where written we can reduce the fraction or not. Even adding 1/2 to 1/3(P1) gives us 5/6 and then subtracting 1/6 from 5/6 gives us P2 = 4/6 = 2/3 which we want.
This can be the correct answer too as we have both the options also given in question.
Either the options should be changed or it should be explicitly written, reduction is not allowed in fractions.

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First	Second
		Add 1/2
		Divide by 3.
		Multiply by 2/3
		Subtract 1/6.
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