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GMAT Club Olympics 2024 (Day 6): In a science experiment, a mouse is

1 2 3 4 5

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15 Jul 2024, 14:29

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Option B

Find P[ (n)(n) ]

A.
Find one of the treat is 1/5
ie possibility are (y)(n)+(n)*(y)=1/5
But we don't know individual probabilities... to calculate
Not Sufficient

B.
(y)(y)=3/10
thus
(n)(n)=1-(y)(y)= 1-3/10=7/10
Sufficient

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Bunuel

In a science experiment, a mouse is placed in a labyrinth with two different treats hidden in it. What is the probability that the mouse will find neither of the treats?

(1) The probability that the mouse will find only one of the two treats is 1/5.
(2) The probability that the mouse will find both treats is 3/10.

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
D. EACH statement ALONE is sufficient to answer the question asked.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

This question was provided by GMAT Club
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The probability here is:

Total (1) = Only one + Both + Neither

Statement 1

Only one = 1/5

1 = 1/5 + Both + Neither

One equation, two unknowns -> cannot solve

-> Statement 1 alone is not enough

Statement 2

Both = 3/10

1 = Only one + 3/10 + neither

One equation, two unknowns -> cannot solve

-> Statement 2 alone is not enough

Both statements combined

1 = 1/5 + 3/10 + neither

-> one equation, one unknown, we can solve for the value

1 - 1/5 - 3/10 = neither

neither = 1/2

Therefore, the answer is C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.

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Let:
- P(A) = Probability that the mouse finds the first treat.
- P(B) = Probability that the mouse finds the second treat.
- P(only one treat) = P(A ∩ ¬B) + P(¬A ∩ B)
- P(both treats) = P(A ∩ B)
- P(neither treat) = P(¬A ∩ ¬B)

From the statements:
1. P(only one treat) = 1/5
2. P(both treats) = 3/10

We need to find P(neither treat).

From probability theory:
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
P(A ∪ B) + P(¬A ∩ ¬B) = 1
P(¬A ∩ ¬B) = 1 - P(A ∪ B)

Step-by-step solution:

1. Statement (1):
P(only one treat) = P(A ∩ ¬B) + P(¬A ∩ B) = 1/5

2. Statement (2):
P(both treats) = P(A ∩ B) = 3/10

3. Using both statements together:
P(A ∪ B) = P(A ∩ ¬B) + P(¬A ∩ B) + P(A ∩ B) = 1/5 + 3/10
P(A ∪ B) = 2/10 + 3/10 = 5/10 = 1/2

4. Therefore:
P(¬A ∩ ¬B) = 1 - P(A ∪ B) = 1 - 1/2 = 1/2

IMO: C

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a = probability that the mouse will find treat one
b = probability that the mouse will find treat two

¿P = (1-a)(1-b) = 1-b-a+ab = 1-(a+b-ab)?

(1)
a(1-b)+b(1-a) = a-ab+b-ab = a+b-2ab = 1/5

There is no way of knowing the value of ab to calculate P

INSUFFICIENT

(2)
ab= 3/10

There is no way of knowing the value of a+b to calculate P

INSUFFICIENT

(1)+(2)
a+b-2ab = 1/5
ab= 3/10

a+b = 1/5 + 6/10 = 4/5

P = 1-(a+b-ab) = 1-(4/5-3/10) = 1-1/2 = 1/2

SUFFICIENT

IMO C

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For sufficiency in this question we either need the probability to get at leas one, where P(0) = 1- P(at least 1), or the probability of finding none.

St1 --> Sufficient: because it gives us the probability of getting at least one, and we can subtract.
St2 --> Insufficient: because we don't know the prob of either getting one or getting none.

Answer A is correct

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P(A)= Mouse takes treat A
P(B)= Mouse takes treat B
1= P(A)+P(B)+P(Both ) +P(Neither)
(A and B are the only 2 possibility in the question )

From Statment 1 - P(only 1 treat )
From Statement 2 : P(takes both )

So Both statements are sufficient

Ans C

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(1) The probability that the mouse will find only one of the two treats is 1/5.
Lets say this is for A, P(A|B) = 1/5 - insufficient

(2) The probability that the mouse will find both treats is 3/10.
P (A B) = 3/10 - insufficient.

Together we can find P(A and B)
Answer (C)

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probability that the mouse will find neither of the treat - (1 - probability the mouse finds both the treats)

(1) The probability that the mouse will find only one of the two treats is 1/5.

NOT SUFFICIENT

(2) The probability that the mouse will find both treats is 3/10.

The answer as per the question: 1-3/10 = 7/10

CORRECT OPTION - B

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Bunuel

This question was provided by GMAT Club
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P(neither)=?
Exacly 1+Both+neither=1

1--Exactly 1=1/5
Not suff we need both also to find neither

2--Both 1=3/10
Not suff we need exacly 1 also to find neither

1+2--- Exactly 1=1/5 Both 1=3/10 So in equation we can find neither so suff

Ans- C Both suff together

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Bunuel

This question was provided by GMAT Club
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Let p(f) & p(s) are respective probabilites.

What is required is 1 - p(f U s) => 1-(p(f)+p(s) - p(f ^ s))
Statement 1 : The probability that the mouse will find only one of the two treats is 1/5.

This meant that p(f)-p(f^s)+p(s)-p(f^s) =1/5

So using this equation 1 - p(f U s) => 1-(p(f)+p(s) - p(f ^ s)) = > 1 -(1/5+p(f^s)).
Hence A & D cannot be a choice.

Statement 2 : The probability that the mouse will find both treats is 3/10.
1 - p(f U s) => 1-(p(f)+p(s) - p(f ^ s)) => 1-(p(f)+p(s) - 3/10).
We cannot conclude any thing with just this info.

Combining stmt 1 & 2 we will be able to find what asked for : 1 -(1/5+p(f^s)).

As we know p(f^s) = 3/10, Over all probability is 1/2;

Hence IMO C.

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Let's define:
P(A) = Probability of finding treat A
P(B) = Probability of finding treat B
P(N) = Probability of finding neither treat (what we're looking for)

From Statement (1): P(only one treat) = 1/5
This means: P(A and not B) + P(B and not A) = 1/5

From Statement (2): P(both treats) = 3/10Statement (1) alone doesn't give us enough information to determine P(N). We need to know either P(both) or P(at least one) to calculate P(N).

Statement (2) alone also doesn't give us enough information. We need to know either P(only one) or P(at least one) to calculate P(N).

Using both statements together: We can use the addition rule of probability: P(at least one) = P(A) + P(B) - P(A and B)

We know: P(A and B) = 3/10 (from statement 2) P(only one) = 1/5 = 2/10 (from statement 1)

So: P(at least one) = P(only one) + P(both) = 2/10 + 3/10 = 5/10 = 1/2

Therefore: P(N) = 1 - P(at least one) = 1 - 1/2 = 1/2

Both statements together are sufficient to answer the question.

The correct answer is C: BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.

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Quote:

We need to find, P(None) = 1 - P(1) - P(2).
(1) P(1) = 1/5. Insuffiecient alone.
(2) P(2) = 3/10. Insufficient alone.

Taking (1) & (2) together can give us P(None) as per the given equation. Hence, both together are sufficient.

The answer is (C)

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Answer: C

P(both) + P(one) + P(none) = 1

P(none) = ?

(1) The probability that the mouse will find only one of the two treats is 1/5.

P(one) = 1/5, but we don't know P(both). Therefore, insufficient.

(2) The probability that the mouse will find both treats is 3/10.

P(both) = 3/10, but we don't know P(one). Therefore, insufficient.

(1) and (2)

P(both) = 3/10, P(one)=1/5. Therefore, sufficient.

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The overall probabiliy of all events $1 = P(1)+ P(2) + P(1+2) + P(0)$, basically meaning that in 100% of the cases the mouse either gets only the first treat, or the second, or both, or none.
The question is: $ P(0) = ?$

[1].
Even if we know $ P(1)+ P(2) = 2*0.2=0.4$, this is insufficient by itself.

[2].
Knowing $P(1+2) = 0.3,$ we can't find P(0). Insufficient.

[1] + [2].
$P(0) = 1 - (P(1)+ P(2)+ P(1+2)) = 1 - 0.4 - 0.3$
This is sufficient.

Therefore, the right answer is C.

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Given: A mouse is placed in a labyrinth with two different treats hidden in it. What is the probability that the mouse will find neither of the treats?

Labyrinth- Will have many turns(left , right) hence finding each event probability will be difficult, unless they specifically provide:

By, General Probability mechanism:

Total probability = 1
Both + Any One + None =1
None =1- Both-Any one

Statement-1
Anyone=1/5, not sufficient alone

Statement-2
Both=3/10, not sufficient alone

Together-1 & 2
None=1-3/10-1/5
None=1-1/2
None=1/2

Sufficient together

Ans-C

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Bunuel

This question was provided by GMAT Club
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Probability of finding neither of the treats is = 1 - probability of finding any one of the treats - probability of finding both the treats
P = 1 - P1 - P2

1) P1 = 1/5, as we don't know P2 we can not get the answer.
Hence, eliminate options A, D.

2) P2 = 3/10, as we don't know P1 we can not get the answer.
Therefore, we can eliminate option B.

Now, if we combine both the statements then we have P1 = 2/10 and P2 = 3/10

=> P = 1 - P1 - P2
=> P = 1 - 2/10 - 3/10
=> P = 1 - 1/2
=> P = 1/2

Answer: C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.

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Treat A x Treat A Total

Treat B m n

x Treat B p q

Total 1

m + n + p + q = 1

Statement 1: $n + p = \frac{1}{5} = 0.2$

=> Insufficient

Statement 2: $m = \frac{3}{10} = 0.3$

=> Insufficient

Combined:

=> m + n + p = 0.2 + 0.3 = 0.5

=> q = 1 - 0.5 = 0.5

=> Sufficient

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C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.

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1- Probability none= Total - {Probability of only 1+Probability of Both}

1- Both is not given
2- Only 1 is not given

Combining 1 and 2 we can solve for Probability none

IMO C

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Bunuel

This question was provided by GMAT Club
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Since the given two treats are placed in a labyrinth and we are nowhere told about the structure of the same, it's safe to assume that the paths to the two treats are distinct, i.e the treats don't have the same probability of being found. But we're given the probability to find either of them, which is the sum of probabilities of finding treat 1 (x) and treat 2 (y) as 1/5, and the probability of finding both, which is the product xy, as 3/10.

We want to determine the probability of the mouse finding no treats, which is simply the complement probability of finding at least one treat, i.e 1-x-y-xy. If the two paths to the two treats were the same, this would have required only one statement, i.e either x+x= 1/5 from I or x.x= 3/10 from II would've both been alone sufficient.

But here,either of the statement alone only tells us of a particular relation between x and y, which are two different variables, thus two equations are required. Combining both the statements we get
1 - (1/5) - (3/10) = 5/10 = 1/2 as our answer.

Thus C is correct.