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In a single Epsilon trial, the probability of Outcome T is 1/4. Supp

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In a single Epsilon trial, the probability of Outcome T is 1/4. Supp  [#permalink]

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In a single Epsilon trial, the probability of Outcome T is 1/4. Suppose a researcher conducts a series of n independent Epsilon trials. Let P = the probability that Outcome T occurs at least once in n trials. Is P > 1/2?

(1) n > 3
(2) n < 6


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Re: In a single Epsilon trial, the probability of Outcome T is 1/4. Supp  [#permalink]

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New post 08 Mar 2015, 15:21
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Bunuel wrote:
In a single Epsilon trial, the probability of Outcome T is 1/4. Suppose a researcher conducts a series of n independent Epsilon trials. Let P = the probability that Outcome T occurs at least once in n trials. Is P > 1/2?

(1) n > 3
(2) n < 6


Kudos for a correct solution.


MAGOOSH OFFICIAL SOLUTION:

So, let’s play with this scenario first a little. Remember, we calculate the “at least” scenario using the complement rule.

Suppose n = 1. Then the probability that T happens at least once (ie. at all!) is 1/4. Of course, this is less than 1/2.

Suppose n = 2.
Attachment:
gdspqop_img2.png
gdspqop_img2.png [ 4.85 KiB | Viewed 2813 times ]

This probability is just less than 1/2.

Suppose n = 3.
Attachment:
gdspqop_img3.png
gdspqop_img3.png [ 4.94 KiB | Viewed 2813 times ]

This probability is greater than 1/2.

Now, think about it. As n increases, there are more and more chances for at least one occurrence of T to happen, so as n increase, the probability of at least one T must also increase. Thus, for all values of n greater than 3, the probability will be even higher, so it must be greater than 1/2.

Statement #1: as we just said, for all larger values of n, at long as we are at n = 3 or higher, the answer to the prompt question will be yes. This statement, alone and by itself, is sufficient.

Statement #2: the problem here, if n < 6, then we could have n = 0 or n = 1, in which case the answer to the prompt question is “no”, or we could have n = 3 or higher, in which case the answer is “yes.” Different possible choices give different answers. This statement, alone and by itself, is insufficient.

First sufficient, second not sufficient.

Answer = A.
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Re: In a single Epsilon trial, the probability of Outcome T is 1/4. Supp  [#permalink]

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New post 06 Mar 2015, 08:15
Probability of at least once is 1 - probability of not happening at all.

n=1>>=1- (1/4)=3/4=0.75
n=3>>=(1- (1/4))^3=0.42
n=4>>=(1- (1/4))^4=.31


statement 1 will have always have values smaller than .5 so sufficient
statement 1 will have values smaller and larger than .5 so insufficient

answer is A
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Re: In a single Epsilon trial, the probability of Outcome T is 1/4. Supp  [#permalink]

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New post 06 Mar 2015, 09:54
Statement 1: P(T at least once in N trials)

case one n=4:

T occurs once . So it will be=1/4*3/4*3/4*3/4= 27/256.

T occurs twice so it will be= 1/4*1/4*3/4*3/4 = 9/256

T occurs three times so it will be= 1/4*1/4*1/4*3/4 = 3/256

T occurs four times so it will be= 1/4*1/4*1/4*1/4 = 1/256

so the sum = 27+9+3+1/256 = 40/256 <1/2 and the answer will be always less than 1/2 as we move to big number

5,6,7,..........so on. Statement 1 is suff

Statement 2: N could be 1,2,3,4,5

case one N=1 T occurs once = 1/4 <1/2

case 2 N=2

T occurs once= 1/4*3/4= 3/16
T occurs twice=1/4*1/4=1/16. So, 1/16+3/16= 4/16 = 1/8<1/2
case 3 N=3
T occurs once= 1/4*3/4*3/4= 9/64
T occurs twice=1/4*1/4*3/4= 3/64
T occurs tree times 1/4*1/4*1/4= 1/64. So, 9+3+1/64= 13/64<1/2
and so on the case for N=4,5 so statment 2 is also suff.
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Re: In a single Epsilon trial, the probability of Outcome T is 1/4. Supp  [#permalink]

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New post 06 Mar 2015, 12:10
Hi 23a2012,

When dealing with these types of probability questions, you have to consider ALL of the various ways that an outcome can occur.

Here, we know that the probability of Outcome T is 1/4 (so the probability of it NOT happening is 3/4)

If we run the experiment 2 times, the probability of Outcome T happening AT LEAST ONCE can be solved in 2 ways:

1) Figure out that probability that it does not happen at all and subtract that from the number 1
2) Figure out ALL of the various ways to get AT LEAST ONE Outcome T.

If you take the second approach, then you have to calculate each of these options:
1) Outcome on the first, NOT on the second.
2) Outcome on the second, NOT on the first.
3) Outcome on the first, Outcome on the second.

In your explanation, the individual calculations that you did were correct, BUT you did NOT consider all of the possibilities, so your overall calculation is incorrect.

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Re: In a single Epsilon trial, the probability of Outcome T is 1/4. Supp  [#permalink]

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New post 07 Mar 2015, 13:51
Bunuel wrote:
In a single Epsilon trial, the probability of Outcome T is 1/4. Suppose a researcher conducts a series of n independent Epsilon trials. Let P = the probability that Outcome T occurs at least once in n trials. Is P > 1/2?

(1) n > 3
(2) n < 6


Kudos for a correct solution.


Use the concept of 1-probability of not obtaining outcome T to account for all possibilities to included the order of success/failure.

Statement 1:
1-probability of not obtaining outcome T
1-(3/4)^4
1-(3/4)^5
The probability is always less than half when n is larger than 3
Sufficient

Statement 2:
when n is less than 6, the probability is less than 1/2 when n is 2 and the probability is greater than 1/2 when n is 3.
Insufficient

Answer: A
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Re: In a single Epsilon trial, the probability of Outcome T is 1/4. Supp  [#permalink]

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Re: In a single Epsilon trial, the probability of Outcome T is 1/4. Supp &nbs [#permalink] 19 Dec 2017, 06:55
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