12 Days of Christmas GMAT Competition 2025 - Day 9: John is applying

Question

John is applying to be the leader of a study group. To qualify, he can either take a short test in which he must solve both of the 2 questions, or he can take a longer test in which he must solve at least 3 of the 4 questions. John’s probability of solving any one question correctly is p, where 0 < p < 1. Would John have a better chance of qualifying if he chose the longer test?

(1) The probability that John qualifies if he chooses the short test is less than 1/4.
(2) The probability that John does not qualify if he chooses the short test is greater than 8/9.

I31-12

truedelulu · Answer

Prob John pass the short test = p^2 Prob John pass the long test = 4C3*p^3*(1-p) + p^4 = p^3(4-3p) For long test to be better chance for John: p^3(4-3p) > p^2 => 4p^3 - 3p^4 - p^2 > 0 => 3p^3 - 3p^4 + p^3 - p^2 > 0 => -3p^3(p-1) + p^2(p-1) > 0 => (p-1)p^2(1-3p) > 0 => 1-3p < 0 (because p-1<0, p^2 > 0) => p > 1/3 (1) p^2 < 1/4 => p < 1/2. This is not sufficient. (2) 1-p^2 > 8/9 => p^2 < 1/9 => p < 1/3. So John definitely doesn't have a better chance with longer test. SUFFICIENT. Answer: B

kapoora10 · Answer

Based on the question the probabilities of short and long question given the probability of john solving a question as p =>

Short (both right), S => p^2
Long (at least 3/4 right), L => 4C3 * p^3 * (1-p) + p^4 = 4p^3(1-p)+p^4

Question: Is 4p^3(1-p) +p^4 > p^2 (L>S)

Statement 1 => P^2 < 1/4 => p<1/2. Hence, john is weak at each question, adding more required questions reduces his chances. Longer is worse. Sufficient.
Statement 2 => 1-p^2 > 8/9 => p<1/3. Puts p well below half, similar to statement 1 => Longer is worse. Sufficient.

Each alone is sufficient => D

remdelectus · Answer

2 probabilities=P^2
(4/3)P^3(1-P)+(4/4)P^4=4P^3(1-P)+P^4=4P^3-3P^4
4P^3-3P^4>P^2
4p^3-3p^4>p^2, p^2(4p-3p^2-1)>0
p^2>0
4p-3p^2-1>0
-3p^2+4p-1>0
3p^2-4p+1<0
(3p-1)(p-1)<0
1/3<p<1
p^2<1/4, p<1/2
p<1/3
statement 1 alone is insufficient
1-p^2>8/9
p^2<1/9, p<1/3
statement 2 alone is sufficient
B. Statement 2 alone is sufficient, but statement 1 alone is not sufficient.

Reon · Answer

John is applying to be the leader of a study group. To qualify, he can either take a short test in which he must solve both of the 2 questions, or he can take a longer test in which he must solve at least 3 of the 4 questions. John’s probability of solving any one question correctly is p, where 0 < p < 1. Would John have a better chance of qualifying if he chose the longer test?
Probability of qualifying short test= p*p= p2
Probability of qualifying long test= (4/3)p3(1-p)1 + (4/4)p4(1-p)0 = p3(4-3p)
We need to find whether John will have a better chance of qualifying if he chose the longer test = p3(4-3p)>p2
3p2-4p+1<0
(3p-1)(p-1)<0
1/3 <p<1
The values of p lies between 1/3 and 1.

(1) The probability that John qualifies if he chooses the short test is less than 1/4.
p2<1/4 or p<1/2
p can take any value less than 1/2 and that value can be anything less or more than 1/3, as multiple values are possible. It is insufficient

(2) The probability that John does not qualify if he chooses the short test is greater than 8/9.
(1-p2)>8/9
p2<1/9 or p<1/3
This gives us a definite answer that p is less than 1/3 and hence he does not have a better chance to with the longer test. It is sufficient.

B

forestmayank · Answer

Probability of correctly solving one ques = p (given)
Therefore, probability of incorrect ans = (1-p)

Probability of qualifying short test with both correct = p x p = p^2

Probability of qualifying long test with
A. 3 correct and 1 incorrect
p x p x p x (1-p) = p^3(1-p)

Now any of the 4 questions can be correct or incorrect. This gives us 4 chances. Therefore, 4 x p^3(1-p)

B. All 4 correct = p x p x p x p = p^4
Combining both

Qualifying long test = 4p^3(1-p) + p^4 = 4p^3 - 3p^4

For long test to have better chances,
4p^3 - 3p^4 > p^2
Dividing by p^2 and solving
3p^2 - 4p + 1 < 0
p > 1/3

From the statements,
Statement 1

John passes short test < 1/4
therefore, p^2 <1/4 OR p < 1/2

This means that p can be >1/3 or <1/3 which is required for longer test to be better.
Hence, Statement 1 alone is not sufficient.

Statement 2
Prob of not qualifying with Short > 8/9
1 - p^2 > 8/9
p^2 < 1/9
p < 1/3

Therefore, longer test is not better.

Hence, Statement 2 alone is sufficient and 1 alone is not sufficient.
Answer, Option B

linnet · Answer

Statement one is not sufficient but statement two alone is sufficient

redandme21 · Answer

short test
2 questions correct p+p=p^2
Probability = p^2

longer test
4 questions correct p*p*p*p = p^4
1 question incorrect 4C1*p*p*p*(1-p) = 4p^3 - 4p^4
Probability = p^4 + 4p^3 - 4p^4 = 4p^3 - 3p^4

Compare probabilities:
4p^3 - 3p^4 > p^2
4p^3 - 3p^4 - p^2 > 0
p^2(4p - 3p^2 - 1) > 0 -> p^2 is always >0
4p - 3p^2 - 1 > 0
3p^2 - 4p + 1 < 0
(3p-1)(p-1) < 0 -> p-1 is always <0
3p - 1 > 0
p>1/3

The question is if p>1/3

(1)
p^2<1/4
p<1/2

if p=0.4 the answer is yes
if p=0.25 the answer is no

Insufficient

(2)
probability(complementary event) = 1-probability(event)
1-p^2>8/9
p^2<1/9
p<1/3

The answer is no

Sufficient

IMO B

dolortempore · Answer

Its good question!

so assume probability of getting answer right as p

now,
P(S) for getting short test success full is = p^2
P(L) = 4p^3 - 3P^2

Now we will check the scenario where
long paper is good as compare to short paper
now P(L) > P(S)

it will give quadratic equation

(p-1/3)(p-1) >0

Means p belongs to (1/3,1) then getting long paper success is high as compare to short and vice versa

Now

statement 1 says

P(S) and qualify < 1/4
p2<1/4
mean
P <1/2

and we know we will be able to comment only if p is <1/3 or >1/3

so if p<1/2 then we cant say if long or short will be great hence insufficient

Statement 2 says

P(S) not qualify > 8/9

1-p2 >8/9
p2<1/9
p<1/3

hence we can say that in this range short paper is good as compare to long hence sufficent

Hence answer (B)

bhanu29 · Answer

Short test: p2p^2 Long test: 4p3−3p44p^3 - 3p^4 Long test is better when p>13p > \frac{1}{3} (1) p2<14⇒p<12p^2 < \frac{1}{4} \Rightarrow p < \frac{1}{2} Could be p=0.4p = 0.4(YES) or p=0.3p = 0.3 (NO) → Insufficient (2) 1−p2>89⇒p<131 - p^2 > \frac{8}{9} \Rightarrow p < \frac{1}{3} Definitively NO → Sufficient Answer: B

Prakruti_Patil · Answer

8/9 is almost 1 and 1 is the highest probability, this leaves him with a clear choice, therefore B) is correct

sanjitscorps18 · Answer

Let 'p' be the probability of John solving a question correctly

Short test (must correctly solve both questions)
To become leader probability required = p*p = p^2

Long test (must correctly solve at least 3 questions)
To become leader probability required = 4C3*p^3(1-p) + p^4
= 4*p^3(1-p) + p^4
=> p^3 (4 - 4p + p)
=> p^3 (4 -3p)

We need to check if the longer test is better, hence
p^3(4-3p) > p^2

Since p is always positive, we get
p(4-3p) > 1
=> 4p - 3p^2 > 1
=> -3p^2 + 4p - 1 > 0
=> 3p^2 - 4p + 1 < 0
=> (3p-1)(p-1) < 0
This implies either of the above is negative
But p-1 has to be negative as p-1 < 0 results in p<1. This is always true
=> 3p - 1 > 0
=> p > 1/3

Hence, if p > 1/3 then the long test is better

1. John qualifies choosing the short test is less than 1/4
This tells that p^2 < 1/4
=> P < 1/2
By this we cannot say whether the short test is better or the long test as we don't know if p is greater than 1/3 or less than 1/3 as both can be below 1/2
Insufficient

2. The probability that John does not qualify if he chooses the short test is greater than 8/9
Not qualifying for the short test = 1 - qualifying for short test
=> Not qualifying > 8/9
=> 1 - p^2 > 8/9
=> p^2 < 1-8/9
=> p^2 < 1/9
=> P < 1/3
This tells us that the long test is not beneficial for John
Sufficient

Option B

Bunuel

John is applying to be the leader of a study group. To qualify, he can either take a short test in which he must solve both of the 2 questions, or he can take a longer test in which he must solve at least 3 of the 4 questions. John’s probability of solving any one question correctly is p, where 0 < p < 1. Would John have a better chance of qualifying if he chose the longer test?

(1) The probability that John qualifies if he chooses the short test is less than 1/4.
(2) The probability that John does not qualify if he chooses the short test is greater than 8/9.

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Mardee · Answer

Let the qualification probability be p

We are given,
John becomes leader of a study group if he,
Takes short test and solves both of the 2 questions
=> P(qualify|short) = p^2
Takes long test and solves atleast 3 out of the 4 questions
=> P(qualify|long) = 4(p^3)(1-p) + p^4 = 4p^3 - 3p^4

We want to know if John has better chance if he chooses the longer test
=> 4p^3 - 3p^4 > p^2
=> p^2 * (4p - 3p^2 - 1) > 0
=> 4p - 3p^2 - 1 > 0 (Since, 0 < p < 1 , p^2 >0 always )
=> 3p^2 - 4p + 1 < 0

=> p = 1/3, 1

=> Long test is seen to be better for -> 1/3 < p < 1 ----------> Z

Now lets go through each of the given statements 1 by 1 to see if John has a better chance of qualifying if he chose the longer test

Statement (1) - The probability that John qualifies if he chooses the short test is less than 1/4

=> p^2 < 1/4
=> p < 1/2
=> 0 < p < 1/2

From what we know above from Z, in this interval, longer test can be better (p=0.4) or worse (p=0.2)

Statement(1) is not sufficient
Statement (2) - The probability that John does not qualify if he chooses the short test is greater than 8/9

=> 1- p^2 > 8/9
=> p < 1/3
=> 0 < p < 1/3

From what we know above from Z, we can say that shorter test is better here since longer test is better if 1/3<p<1

Statement(2) alone is sufficient

B. Statement (2) alone is sufficient, but statement(1) alone is not sufficient

geocircle · Answer

Probability of pass the short test = p^2
Probability of pass the long test = p^4 + 4p^3(1-p)

Is p^4 + 4p^3(1-p) > p^2?
p^4 + 4p^3 - 4p^4 > p^2
-3p^4 + 4p^3 - p^2 > 0

Divide by p^2 (positive):
-3p^2 + 4p - 1 > 0

Factorize getting the roots:
-3(p-1)(p-1/3) > 0
(p-1)(p-1/3) < 0

0<p<1 so p-1<0
p-1/3 > 0
p > 1/3 -> answer to this

(1)
p^2 < 1/4
p < 1/2

p can be >1/3 or <1/3

Condition insufficient

(2)
Probability of not pass the short test = 1 - Probability of pass the short test = 1 - p^2

1-p^2 > 8/9
p^2 < 1/9
p < 1/3

p is not >1/3 -> answer no

Condition sufficient

Answer B

msignatius · Answer

There are few things we can do with the stem.

John needs 100% in the two-question test to qualify.

OR

John needs 75% in the four-question test to qualify.

It's a tricky choice when you consider that the probability of solving each question is between 0 and 100% (as given by 0 < p < 1).

If the probability was, for instance, 90%, then solving 2 questions correctly would've led to a success-odds of 90*90 / 100*100 = 81%. And for the 4 question-er, at 90% per question, we need any 3 right or 90*90*90 / 100*100*100 = 72.9%, but since we can get both all 4 right and 3 right / 1 wrong, we get the additional benefit of adding the probabilities here. For 4 right at this rate, the odds will be 0.9^4 or ~0.65; and for 3 right and 1 wrong, the probability will be 0.9^3*0.1 (for the 10% probability of getting an answer wrong), which is around 7.29%. But remember, any 3 of the 4 questions can be right, hence 4! / 3! = 4*7.29 =  around 29% will be our odds, which if we add to 65%, will give us effective odds of around 94%

Hence, we really need to figure out relationships between the probability, and hence, we need to see the statements.

Statement 1: The probability that John qualifies if he picks the short test is less than 1/4.

Okay, so, less than 25%, will mean less than x^2 = 0.25, effectively, which is achievable if the odds of success for each question is less than 50% (x = <0.5)

At this rate, we get odds of 0.5^4 = 0.0625 or 6.25% for all 4 correct, and (0.5^3)*0.5 = 6.25% for 3/4 correct and 1 wrong (which at 4!/3! instances, which be 25%) - so we have the odds of 31.25% for qualifying via the long test, which is higher than the odds of John qualifying with the short test.

But there's certainly one pattern you'll be able to make out: the lower we take the probability (less than 25, to, say 10%), the more the exponents will diminish the probabilities. So, when we're comparing x^2 vs x^4, as we do with the longer test, we might reach a case where the longer test is less probable for success.

Let's take the 0.1 or 10.5% I've mentioned; to reach a 10% success rate, x^2 = 0.1, which means for each question, we'll have around 30% as the success rate, or 0.3.

For 4 correct, now, the probability will be 0.3^4, which is less than 1%, and for 3 correct and 1 wrong, the probability 0.33^3*0.66 will be around 2%, and in 4 instances, that will give us as a success  rate of ~9% longer test. This is a bit lower than the odds of the 10% we assumed for the shorter test.

Hence, we get conflicting answers.  Hence, Statement I is not enough.

Statement 2: The probability that John won't qualify if he picks the shorter test is greater than 8/9.

I think it's pretty well established that if the probability of success with the shorter-test keeps on going down, the probability with the longer-test will go down even more.

We can find the result with the probability of success being lesser than 1/9 for the shorter test (which the statement essentially gives) by assuming 1/9, and if in this we get an identical or lower probability of success for the longer test, we have our answer.

First, we find the individual probability of each question: x^2 = 1/9; hence x = 1/3.

Now, 1/3^4 = 1/81 is the probability for when all 4 questions are correct; and (1/27)*2/3 = 2 / 81 (4 times over = 8 / 81) is the probability we have when 3 are correct, and 1 is wrong. Adding this to 1 / 81, we get 9/81, which is just 1/9.

Hence, anything less than 1/9 probability for success with the shorter-test will still prove the shorter-test more probable for John's qualification. This gives us a definitive answer, hence, B alone is sufficient.

topgmat25 · Answer

Short test:
p^2 (both correct)

Long test:
p^4 (4 correct)
p^3(1-p) (3 correct), but the incorrect can be any of the 4, so 4p^3(1-p).
total: p^4+4p^3(1-p) = p^4+4p^3-4p^4 = 4p^3-3p^4

The question to answer is:
p^2 < 4p^3-3p^4
4p^3-3p^4-p^2>0
p^2(4p-3p^2-1)>0
-3p^2(p^2-4p/3+1/3)>0
-3p^2(p-1)(p-1/3)>0

-3p^2(p-1) is > 0 if 0<p<1

p-1/3>0
p>1/3

Is p is greater than 1/3?

(1)
p^2<1/4 implies that p<1/2=0.5

If p=0.45>1/3 then the answer is yes
If p=0.1<1/3 then the answer is no

Condition (1) is insufficient

(2)
"Does not qualify" implies the complementary even.

The probability of qualify is less than:
1-8/9=1/9

p^2<1/9 implies that p<1/3

the answer is no

Condition (2) is sufficient

The answer is B

rahumangal · Answer

prob that john wins both test = p^2 prob that john wins 3or 4= 4c3 p^3(1-p) + p^4=p^3 (4-3p) for long test to be better than short test p^3(4-3p)> p^2 (3p-1)(p-1)<0 1/38/9 p<1/3 sufficient Ans=B

MANASH94 · Answer

Let probability of solving one question correctly = P(short) = p^2 P(long) = 4(p^3)(1 − p) + p^4 Comparing chances: P(long) > P(short) 4(p^3)(1 − p) + p^4 > p^2 Factor p^2: p^2(4p(1 − p) + p^2) > p^2 Since p > 0, dividing both sides by p^2: 4p(1 − p) + p^2 > 1 Expand: 4p − 4p^2 + p^2 > 1 4p − 3p^2 > 1 Rearranging: 3p^2 − 4p + 1 < 0 Factoring: (3p − 1)(p − 1) < 0 So: p > (1/3) Checking the statements (1) P(short) < 1/4 p^2 < (1/2)^2 p < 1/2 This allows p = 2/5 and p = 1/5 : long test sometimes better, sometimes worse. Not sufficient Statement (2): 1 − p^2 > 8/9 p^2 < 1/9 = (1/3)^2 p < 1/3 Thus: P(short) > P(long) Sufficient Answer should be B alone is sufficient.

prepapr · Answer

probability of solving both questions in short test = p^2 probability of solving atleast 3 of 4 questions in long test = 4C3 * (p^3) (1-p) + (p^4) = 4p^3 - 3p^4 To have a better chance of qualifying in long test 4p^3 - 3p^4 > p^2 p^2 * (4p - 3p^2 - 1) >0 since p^2 > 0 we get 4p - 3p^2 - 1 >0 3p^2 - 4p + 1 <0 Solving this, we get p = 1/3, 1 Thus long test is better if 1/3 < p < 1 Analysing statements A) P short < 1/4 p^2 < 1/4 p < 1/2 0 8/9 p<1/3 This means short test is better, answer is No This is sufficient. Hence answer is B

firefox300 · Answer

The probability in the short test is p*p = p^2
The probability in the longer test is the sum of the probability(exactly 3 correct) plus probability(exactly 4 correct) = 4C3*p^3(1-p) + 4C4*p^4 = 4p^3 - 3p^4

Compare the two probabilities:
4p^3 - 3p^4 > p^2
3p^4 - 4p^3 + p^2 < 0
3p^2 - 4p + 1 < 0
p^2 - 4p/3 + 1/3 < 0
(p-1/3)(p-1) < 0

0<p<1 -> -1<p-1<0 -> p-1<0
p-1/3 > 0
p > 1/3?

(1)
p^2 < 1/4 if p < 1/2
This includes p > 1/3 and p <= 1/3

Insufficient

(2)
P(does not qualify) = 1 - P(qualify) = 1-p^2
1-p^2>8/9
p^2<1/9
p<1/3

John wouldn't have a better chance of qualifying if he chose the longer test.

Sufficient

The correct answer is B

canopyinthecity · Answer

Let p be probability of qualifying the test

P(short test) = p^2
P(longer test) = 4C3 * p^3 * (1-p) + p^4

For both options individually 
Since 0<p<1, p^3<p^2<p

Which implies longer lest has less probability than shorter test. Hence both statements are individually enough to answer the question.

Hence (D) is the answer

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