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A 5 meter long wire is cut into two pieces. If the longer [#permalink]

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17 Dec 2010, 08:23

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A 5 meter long wire is cut into two pieces. If the longer piece is then used to form a perimeter of a square, what is the probability that the area of the square will be more than 1 if the original wire was cut at an arbitrary point?

The area of the square will be more than 1 if and only if the longer piece of the wire is longer than 4. To produce such a result, the cutting point has to be either on the first meter of the wire or on its last meter. The probability of this is 2/5 .

I have a question. Since area of the square has to be more than 1, its sides have to be more than 1. If its sides are more than 1, its perimeter will be more than 4. In order for this to happen, a piece larger than 4 meters has to be cut. The smallest piece need would be 4 meters and 1 centimeter (or millimeter for that matter). While I agree that the chance of wire being cut on any of the first 1 meters of it is 2/5, don't we need to calculate the probability of it being cut (at least) at 4 meters and 1 centimeter (or mm)? If so, the probability that it will get cut within first 99 cm is 2*99/500, which comes out to 99/250. This is close to 2/5 yes, but isn't this more correct way to look at it?

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Last edited by Vyshak on 14 Apr 2016, 19:41, edited 1 time in total.

A 5 meter long wire is cut into two pieces. If the longer piece is then used to form a perimeter of a square, what is the probability that the area of the square will be more than 1 if the original wire was cut at an arbitrary point?

A) 1/6 B) 1/5 C) 3/10 D) 1/3 E) 2/5

OA says it is 2/5 because:

The area of the square will be more than 1 if and only if the longer piece of the wire is longer than 4. To produce such a result, the cutting point has to be either on the first meter of the wire or on its last meter. The probability of this is 2/5 .

I have a question. Since area of the square has to be more than 1, its sides have to be more than 1. If its sides are more than 1, its perimeter will be more than 4. In order for this to happen, a piece larger than 4 meters has to be cut. The smallest piece need would be 4 meters and 1 centimeter (or millimeter for that matter). While I agree that the chance of wire being cut on any of the first 1 meters of it is 2/5, don't we need to calculate the probability of it being cut (at least) at 4 meters and 1 centimeter (or mm)? If so, the probability that it will get cut within first 99 cm is 2*99/500, which comes out to 99/250. This is close to 2/5 yes, but isn't this more correct way to look at it?

Basically you are saying that the probability should be almost 2/5 but not exactly 2/5, because wire can be cut at 1 or 4 meters exactly and in this case probability will be a little less than 2/5. But this is not true, 1 and 4 meters marks are points and point by definition has no length or any other dimension.

It's similar to the followoing example: the probability that a number X picked from the range (0,5) is more than 4 is 1/5 as well as the probability that a number X picked from the range (0,5) is more than or equal to 4 is also 1/5.

Re: A 5 meter long wire is cut into two pieces. If the longer [#permalink]

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17 Dec 2010, 12:43

Bunuel wrote:

MisterEko wrote:

A 5 meter long wire is cut into two pieces. If the longer piece is then used to form a perimeter of a square, what is the probability that the area of the square will be more than 1 if the original wire was cut at an arbitrary point?

A) 1/6 B) 1/5 C) 3/10 D) 1/3 E) 2/5

OA says it is 2/5 because:

The area of the square will be more than 1 if and only if the longer piece of the wire is longer than 4. To produce such a result, the cutting point has to be either on the first meter of the wire or on its last meter. The probability of this is 2/5 .

I have a question. Since area of the square has to be more than 1, its sides have to be more than 1. If its sides are more than 1, its perimeter will be more than 4. In order for this to happen, a piece larger than 4 meters has to be cut. The smallest piece need would be 4 meters and 1 centimeter (or millimeter for that matter). While I agree that the chance of wire being cut on any of the first 1 meters of it is 2/5, don't we need to calculate the probability of it being cut (at least) at 4 meters and 1 centimeter (or mm)? If so, the probability that it will get cut within first 99 cm is 2*99/500, which comes out to 99/250. This is close to 2/5 yes, but isn't this more correct way to look at it?

Basically you are saying that the probability should be almost 2/5 but not exactly 2/5, because wire can be cut at 1 or 4 meters exactly and in this case probability will be a little less than 2/5. But this is not true, 1 and 4 meters marks are points and point by definition has no length or any other dimension.

It's similar to the followoing example: the probability that a number X picked from the range (0,5) is more than 4 is 1/5 as well as the probability that a number X picked from the range (0,5) is more than or equal to 4 is also 1/5.

Hope it's clear.

Hm, not sure I understood you (or that you understood my question). What I said is that since the piece cut has to be bigger than 4, even marginally bigger, probability will be slightly less than 2/5. Say we can only count in cm. Wire is 500 cm long. In order for square to have area more than 1 meter, perimeter needs to be at least slightly bigger than 400 cm. In that case, the next least place that wire needs to be cut on (and satisfy the area of a square being bigger than 1) would be on 99th cm (down to the 1st cm) or 401st cm (up to 499). Either way, there are 99 positions on the first meter and 99 on the second one that satisfy the length needed. Since the total is 500 cm, probability would be 2*99/500 or 99/250.

Please, forgive me if I am being nuisance, I am kinda intrigued by this.
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Hm, not sure I understood you (or that you understood my question). What I said is that since the piece cut has to be bigger than 4, even marginally bigger, probability will be slightly less than 2/5. Say we can only count in cm. Wire is 500 cm long. In order for square to have area more than 1 meter, perimeter needs to be at least slightly bigger than 400 cm. In that case, the next least place that wire needs to be cut on (and satisfy the area of a square being bigger than 1) would be on 99th cm (down to the 1st cm) or 401st cm (up to 499). Either way, there are 99 positions on the first meter and 99 on the second one that satisfy the length needed. Since the total is 500 cm, probability would be 2*99/500 or 99/250.

Please, forgive me if I am being nuisance, I am kinda intrigued by this.

OK, let me ask you this: what is the probability that the wire will be cut so that we get pieces of exactly 4m and 1m (so at 1m or at 4m)?

What I'm saying is that the probability that the length of a longer piece will be more than 4 OR more than or equal to 4 is the same and equal to 2/5.
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Re: A 5 meter long wire is cut into two pieces. If the longer [#permalink]

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17 Dec 2010, 13:03

Bunuel wrote:

MisterEko wrote:

Hm, not sure I understood you (or that you understood my question). What I said is that since the piece cut has to be bigger than 4, even marginally bigger, probability will be slightly less than 2/5. Say we can only count in cm. Wire is 500 cm long. In order for square to have area more than 1 meter, perimeter needs to be at least slightly bigger than 400 cm. In that case, the next least place that wire needs to be cut on (and satisfy the area of a square being bigger than 1) would be on 99th cm (down to the 1st cm) or 401st cm (up to 499). Either way, there are 99 positions on the first meter and 99 on the second one that satisfy the length needed. Since the total is 500 cm, probability would be 2*99/500 or 99/250.

Please, forgive me if I am being nuisance, I am kinda intrigued by this.

OK, let me ask you this: what is the probability that the wire will be cut so that we get pieces of exactly 4m and 1m (so at 1m or at 4m)?

What I'm saying is that the probability that the length of a longer piece will be more than 4 OR more than or equal to 4 is the same and equal to 2/5.

Now I get it. Thank you... Probabilities will always get ya'... Cheers!
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[highlight]Monster collection of Verbal questions (RC, CR, and SC)[/highlight] http://gmatclub.com/forum/massive-collection-of-verbal-questions-sc-rc-and-cr-106195.html#p832142

[highlight]Massive collection of thousands of Data Sufficiency and Problem Solving questions and answers:[/highlight] http://gmatclub.com/forum/1001-ds-questions-file-106193.html#p832133

Re: A 5 meter long wire is cut into two pieces. If the longer [#permalink]

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06 Nov 2013, 16:06

Bunuel wrote:

MisterEko wrote:

A 5 meter long wire is cut into two pieces. If the longer piece is then used to form a perimeter of a square, what is the probability that the area of the square will be more than 1 if the original wire was cut at an arbitrary point?

A) 1/6 B) 1/5 C) 3/10 D) 1/3 E) 2/5

OA says it is 2/5 because:

The area of the square will be more than 1 if and only if the longer piece of the wire is longer than 4. To produce such a result, the cutting point has to be either on the first meter of the wire or on its last meter. The probability of this is 2/5 .

I have a question. Since area of the square has to be more than 1, its sides have to be more than 1. If its sides are more than 1, its perimeter will be more than 4. In order for this to happen, a piece larger than 4 meters has to be cut. The smallest piece need would be 4 meters and 1 centimeter (or millimeter for that matter). While I agree that the chance of wire being cut on any of the first 1 meters of it is 2/5, don't we need to calculate the probability of it being cut (at least) at 4 meters and 1 centimeter (or mm)? If so, the probability that it will get cut within first 99 cm is 2*99/500, which comes out to 99/250. This is close to 2/5 yes, but isn't this more correct way to look at it?

Basically you are saying that the probability should be almost 2/5 but not exactly 2/5, because wire can be cut at 1 or 4 meters exactly and in this case probability will be a little less than 2/5. But this is not true, 1 and 4 meters marks are points and point by definition has no length or any other dimension.

It's similar to the followoing example: the probability that a number X picked from the range (0,5) is more than 4 is 1/5 as well as the probability that a number X picked from the range (0,5) is more than or equal to 4 is also 1/5.

Hope it's clear.

Hey Bunuel,

I'm having hard time understanding your example: "the probability that a number X picked from the range (0,5) is more than 4 is 1/5 as well as the probability that a number X picked from the range (0,5) is more than or equal to 4 is also 1/5." Basically, I'm getting confused because I think the probability of picking a number from the range (0,5), which I converted to a set {0,1,2,3,4,5}, is 1/6 (6 = total number of terms). So probability of picking a number > 4 is 1/6 (because the only possibility is 5 from the set). But probability of picking up a number >=4 is 2/6 because now two outcomes, 4 and 5, can be considered successful. So, can you kindly let me know where I'm going wrong? Also, is the probability in the rope = Length of a unit of rope/ Total Length of the rope? Appreciate your help.. Thanks

A 5 meter long wire is cut into two pieces. If the longer piece is then used to form a perimeter of a square, what is the probability that the area of the square will be more than 1 if the original wire was cut at an arbitrary point?

A) 1/6 B) 1/5 C) 3/10 D) 1/3 E) 2/5

OA says it is 2/5 because:

The area of the square will be more than 1 if and only if the longer piece of the wire is longer than 4. To produce such a result, the cutting point has to be either on the first meter of the wire or on its last meter. The probability of this is 2/5 .

I have a question. Since area of the square has to be more than 1, its sides have to be more than 1. If its sides are more than 1, its perimeter will be more than 4. In order for this to happen, a piece larger than 4 meters has to be cut. The smallest piece need would be 4 meters and 1 centimeter (or millimeter for that matter). While I agree that the chance of wire being cut on any of the first 1 meters of it is 2/5, don't we need to calculate the probability of it being cut (at least) at 4 meters and 1 centimeter (or mm)? If so, the probability that it will get cut within first 99 cm is 2*99/500, which comes out to 99/250. This is close to 2/5 yes, but isn't this more correct way to look at it?

Basically you are saying that the probability should be almost 2/5 but not exactly 2/5, because wire can be cut at 1 or 4 meters exactly and in this case probability will be a little less than 2/5. But this is not true, 1 and 4 meters marks are points and point by definition has no length or any other dimension.

It's similar to the followoing example: the probability that a number X picked from the range (0,5) is more than 4 is 1/5 as well as the probability that a number X picked from the range (0,5) is more than or equal to 4 is also 1/5.

Hope it's clear.

Hey Bunuel,

I'm having hard time understanding your example: "the probability that a number X picked from the range (0,5) is more than 4 is 1/5 as well as the probability that a number X picked from the range (0,5) is more than or equal to 4 is also 1/5." Basically, I'm getting confused because I think the probability of picking a number from the range (0,5), which I converted to a set {0,1,2,3,4,5}, is 1/6 (6 = total number of terms). So probability of picking a number > 4 is 1/6 (because the only possibility is 5 from the set). But probability of picking up a number >=4 is 2/6 because now two outcomes, 4 and 5, can be considered successful. So, can you kindly let me know where I'm going wrong? Also, is the probability in the rope = Length of a unit of rope/ Total Length of the rope? Appreciate your help.. Thanks

Why do you consider only integers? The numbers from 0 to 5 consists of ALL numbers from 0 to 5, not only of integers.
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Re: A 5 meter long wire is cut into two pieces. If the longer [#permalink]

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07 Nov 2013, 10:14

Hey Bunuel,

I'm having hard time understanding your example: "the probability that a number X picked from the range (0,5) is more than 4 is 1/5 as well as the probability that a number X picked from the range (0,5) is more than or equal to 4 is also 1/5." Basically, I'm getting confused because I think the probability of picking a number from the range (0,5), which I converted to a set {0,1,2,3,4,5}, is 1/6 (6 = total number of terms). So probability of picking a number > 4 is 1/6 (because the only possibility is 5 from the set). But probability of picking up a number >=4 is 2/6 because now two outcomes, 4 and 5, can be considered successful. So, can you kindly let me know where I'm going wrong? Also, is the probability in the rope = Length of a unit of rope/ Total Length of the rope? Appreciate your help.. Thanks[/quote]

Why do you consider only integers? The numbers from 0 to 5 consists of ALL numbers from 0 to 5, not only of integers.[/quote]

Thanks for your prompt help. But I'm still confused Bunuel! There are infinite real numbers between 0 & 5, so how did we get 1/5 as the probability? I'm unable to visualize this problem. Can you kindly explain it to me in terms of successful outcomes/total outcomes? Thanks for your help...

Thanks for your prompt help. But I'm still confused Bunuel! There are infinite real numbers between 0 & 5, so how did we get 1/5 as the probability? I'm unable to visualize this problem. Can you kindly explain it to me in terms of successful outcomes/total outcomes? Thanks for your help...

Visualization is exactly what should help. Consider a number line: {total} is line segment of 5 units (from 0 to 5) and {favorable} is a line segment of 1 unit (from 4 to 5), thus P = {favorable}/{total} = 1/5.

Re: A 5 meter long wire is cut into two pieces. If the longer [#permalink]

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09 Nov 2013, 12:54

Bunuel wrote:

prsnt11 wrote:

Thanks for your prompt help. But I'm still confused Bunuel! There are infinite real numbers between 0 & 5, so how did we get 1/5 as the probability? I'm unable to visualize this problem. Can you kindly explain it to me in terms of successful outcomes/total outcomes? Thanks for your help...

Visualization is exactly what should help. Consider a number line: {total} is line segment of 5 units (from 0 to 5) and {favorable} is a line segment of 1 unit (from 4 to 5), thus P = {favorable}/{total} = 1/5.

Hope it's clear.

Hi,

Two comments: 1- Can sbdy please edit the question so that OA is not spoiled? 2- I don't agree with OA = 2/5, since question says that we use the longer side to create the square. Therefore no distinction is made between the "two" sides of the rope. Only the longer side is considered for our purpose. What I am saying is that I would pick 1/5 as the answer.

Hm, not sure I understood you (or that you understood my question). What I said is that since the piece cut has to be bigger than 4, even marginally bigger, probability will be slightly less than 2/5. Say we can only count in cm. Wire is 500 cm long. In order for square to have area more than 1 meter, perimeter needs to be at least slightly bigger than 400 cm. In that case, the next least place that wire needs to be cut on (and satisfy the area of a square being bigger than 1) would be on 99th cm (down to the 1st cm) or 401st cm (up to 499). Either way, there are 99 positions on the first meter and 99 on the second one that satisfy the length needed. Since the total is 500 cm, probability would be 2*99/500 or 99/250.

Please, forgive me if I am being nuisance, I am kinda intrigued by this.

OK, let me ask you this: what is the probability that the wire will be cut so that we get pieces of exactly 4m and 1m (so at 1m or at 4m)?

What I'm saying is that the probability that the length of a longer piece will be more than 4 OR more than or equal to 4 is the same and equal to 2/5.

I think the answer 2/5 is flawed. Let us take five points 1, 2, 3, 4 and 5. For the longer wire to be more than or equal to 4m, 2 cases are there, it has to be cut at 1m or before OR 4m or after.

Let us take the first case. For example if it is cut at 1m, the longer wire would be 4m. Assume that each 1m is divided into cms. So there are 99 points out of a total of 500 points when the longer wire can be more than 4 m. for example if it is cut at 99th cm the longer wire would be 401 cm or > 4m. and so on. So there are totally 100 points including the exact 1m point out of a total of 500 points. So the probability that the longer wire is equal to or more than 4m is 100/500=1/5. The other case is wire cut at 4m and above. Each case has a probability of 1/5 so that the overall probability is 2/5.

The above is also equivalent to saying that the shorter wire is <= 1m. That is the probability is 2/5 when the longer wire is 4m and the shorter wire is 1m and say when the longer wire is 4.01 m and the shorter wire is 0.99m and so on . But saying equal to or more than 4m is not equivalent to saying less than 1m but only equivalent to saying less than or equal to 1m.

So when we have the shorter wire to be less than 1m, we have to consider only the cases > 4m. When it has to be more than 4m, the probability will be less than the above or less than 2/5.
_________________

Hm, not sure I understood you (or that you understood my question). What I said is that since the piece cut has to be bigger than 4, even marginally bigger, probability will be slightly less than 2/5. Say we can only count in cm. Wire is 500 cm long. In order for square to have area more than 1 meter, perimeter needs to be at least slightly bigger than 400 cm. In that case, the next least place that wire needs to be cut on (and satisfy the area of a square being bigger than 1) would be on 99th cm (down to the 1st cm) or 401st cm (up to 499). Either way, there are 99 positions on the first meter and 99 on the second one that satisfy the length needed. Since the total is 500 cm, probability would be 2*99/500 or 99/250.

Please, forgive me if I am being nuisance, I am kinda intrigued by this.

OK, let me ask you this: what is the probability that the wire will be cut so that we get pieces of exactly 4m and 1m (so at 1m or at 4m)?

What I'm saying is that the probability that the length of a longer piece will be more than 4 OR more than or equal to 4 is the same and equal to 2/5.

I think the answer 2/5 is flawed. Let us take five points 1, 2, 3, 4 and 5. For the longer wire to be more than or equal to 4m, 2 cases are there, it has to be cut at 1m or before OR 4m or after.

Let us take the first case. For example if it is cut at 1m, the longer wire would be 4m. Assume that each 1m is divided into cms. So there are 99 points out of a total of 500 points when the longer wire can be more than 4 m. for example if it is cut at 99th cm the longer wire would be 401 cm or > 4m. and so on. So there are totally 100 points including the exact 1m point out of a total of 500 points. So the probability that the longer wire is equal to or more than 4m is 100/500=1/5. The other case is wire cut at 4m and above. Each case has a probability of 1/5 so that the overall probability is 2/5.

The above is also equivalent to saying that the shorter wire is <= 1m. That is the probability is 2/5 when the longer wire is 4m and the shorter wire is 1m and say when the longer wire is 4.01 m and the shorter wire is 0.99m and so on . But saying equal to or more than 4m is not equivalent to saying less than 1m but only equivalent to saying less than or equal to 1m.

So when we have the shorter wire to be less than 1m, we have to consider only the cases > 4m. When it has to be more than 4m, the probability will be less than the above or less than 2/5.

The correct answer is E (2/5).

A 5 meter long wire is cut into two pieces. If the longer piece is then used to form a perimeter of a square, what is the probability that the area of the square will be more than 1 if the original wire was cut at an arbitrary point?

A. 1/6 B. 1/5 C. 3/10 D. 1/3 E. 2/5

In order the area of a square to be more than 1, its side must be more than 1, or the perimeter must be more than 4. Which means that the longer piece must be more than 4. Look at the diagram below:

-----

If the wire will be cut anywhere at the red region then the rest of the wire (longer piece) will be more than 4 meter long. The probability of that is 2/5 (2 red pieces out of 5).

Re: A 5 meter long wire is cut into two pieces. If the longer [#permalink]

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26 Nov 2013, 18:03

A 5 meter long wire is cut into two pieces. If the longer piece is then used to form a perimeter of a square, what is the probability that the area of the square will be more than 1 if the original wire was cut at an arbitrary point?

That the area of the square is more than 1 square meter means that the perimeter of the square is more than 4 meter. Imagine the wire is divided into 5 pieces: 0__1__2__3__4__5

I see that if we cut the wire at any point from 0 to 1 or any point from 4 to 5, we will have a long wire whose perimeter is more than 4 meter. If we cut the wire at any point from 1 to 4, we get a long wire whose perimeter is less than 4 meter.

Undoubtedly, we have 3 choices if we cut the wire: from 0 to 1, from 1 to 4, and from 4 to 5. Following this reasoning, I think the probability that the area of the square will be more than 1 if the original wire was cut at an arbitrary point should be 2/3.

A 5 meter long wire is cut into two pieces. If the longer piece is then used to form a perimeter of a square, what is the probability that the area of the square will be more than 1 if the original wire was cut at an arbitrary point?

That the area of the square is more than 1 square meter means that the perimeter of the square is more than 4 meter. Imagine the wire is divided into 5 pieces: 0__1__2__3__4__5

I see that if we cut the wire at any point from 0 to 1 or any point from 4 to 5, we will have a long wire whose perimeter is more than 4 meter. If we cut the wire at any point from 1 to 4, we get a long wire whose perimeter is less than 4 meter.

Undoubtedly, we have 3 choices if we cut the wire: from 0 to 1, from 1 to 4, and from 4 to 5. Following this reasoning, I think the probability that the area of the square will be more than 1 if the original wire was cut at an arbitrary point should be 2/3.

Please explain what is wrong with my explanation?

Merging similar topics. Please search before posting.
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Re: A 5 meter long wire is cut into two pieces. If the longer [#permalink]

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24 Aug 2016, 21:58

MisterEko wrote:

A 5 meter long wire is cut into two pieces. If the longer piece is then used to form a perimeter of a square, what is the probability that the area of the square will be more than 1 if the original wire was cut at an arbitrary point?

A) 1/6 B) 1/5 C) 3/10 D) 1/3 E) 2/5

Let the place where the rope is cut be \(x\) and it's used to form a square with \(x\) as perimeter and hence side is \(\frac{x}{4}\) and Area =\(\frac{x^2}{4}\) The area should be more than 1 so \(x^2/4 > 1\) , so \(x^2 > 4\) giving \(x > 2\). Out the places where it can be cut, 1,2,3,4 , 3 and 4 are above 2, so \(\frac{2}{5}\) Answer is E