A 5 meter long wire is cut into two pieces. If the longer piece is the

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.

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!

Thanks Bunuel, thats very good explanation, but could you clarify onemore thing, isn't 2/5 the probability of getting area of the square which equals to 1. Since cutting on any red sides will leave us longer wire with length 4 which is not more than 1 in area but just equal.
Thanks.

Hi Bunnel,

One query...

Seeing the figure how can we say that three(Black dashes) sum to length more than 4?

I think you both do not understand one thing: each dash is 1 meter long. So, again if we cut anywhere at the red region (from 0 to 1 meter or from 4 to 5 meters) then the rest of the wire (longer piece) will be more than 4 meter long.

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.

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...

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.

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.

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:

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?

Your explanation is close, but take another look at your suggested cuts.

If we divide the potential cuts into 0 to 1, 1 to 4, and 4 to 5, we have a 1:3:1 ratio where either end will give us the square with area >1.

Add up the parts to get 5, 2 of which are successful outcomes, and you get a probability of 2/5.

I would very much appreciate if someone can point out the flaw in my logic

the line is cut into 2 pieces ....x , 5-x

assuming x is the larger piece then x>2.5....... ( the larger piece)

side of square = x/4 , area of square = x^2/16 , question is to test x^2/16>1

we get ............-4................4.............. the inequality is true in the ranges x<-4 and x>4 but we have a restriction that x>2.5 thus x>4

thus 4<x<5 , thus probability = (5-4)/5 = 1/5 ............Bunuel where am i going wrong if you plz.

What if 5-x is larger piece and not x. In this case you'd have (5 - x)^2/16 > 1 giving x < 1.

Why do we consider the 1st and 2d red region as two separate instances. the rope doesn't have two distinct ends right? Cut at region 1 or region 2 it gives the same result.