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A frog makes 3 jumps, each exactly 1 meter long. The directions of the

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Bunuel
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A frog makes 3 jumps, each exactly 1 meter long. The directions of the [#permalink] New post   31 Mar 2019, 21:13
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A frog makes 3 jumps, each exactly 1 meter long. The directions of the jumps are chosen independently at random. What is the probability that the frog's final position is no more than 1 meter from its starting position?

A. 1/6
B. 1/5
C. 1/4
D. 1/3
E. 1/2
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C
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A frog makes 3 jumps, each exactly 1 meter long. The directions of the [#permalink] New post   31 Mar 2019, 21:52
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Bunuel wrote:
A frog makes 3 jumps, each exactly 1 meter long. The directions of the jumps are chosen independently at random. What is the probability that the frog's final position is no more than 1 meter from its starting position?

A. 1/6
B. 1/5
C. 1/4
D. 1/3
E. 1/2


Visualization can make this problem a very easy one..

The catch word is ANY direction
We can infer that he would jump on the perimeter of a circle of radius 1 around it.

See the attached figure.
After taking the first jump on the circle say at point A as shown. The next two jumps can take the frog 2 m away from A.
Thus the Area of bigger circle, shaded in the sketch, is the total area where the frog can be, while the area of the smaller circle is the area where the frog is 1m from his initial position.
Required probability = \(\frac{\pi*1^2}{\pi*2^2}=\frac{1}{4}\)

C
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A frog makes 3 jumps, each exactly 1 meter long. The directions of the [#permalink] New post   18 Oct 2020, 22:53
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chetan2u wrote:
Bunuel wrote:
A frog makes 3 jumps, each exactly 1 meter long. The directions of the jumps are chosen independently at random. What is the probability that the frog's final position is no more than 1 meter from its starting position?

A. 1/6
B. 1/5
C. 1/4
D. 1/3
E. 1/2


Visualization can make this problem a very easy one..

The catch word is ANY direction
We can infer that he would jump on the perimeter of a circle of radius 1 around it.

See the attached figure.
After taking the first jump on the circle say at point A as shown. The next two jumps can take the frog 2 m away from A.
Thus the Area of bigger circle, shaded in the sketch, is the total area where the frog can be, while the area of the smaller circle is the area where the frog is 1m from his initial position.
Required probability = \(\frac{\pi*1^2}{\pi*2^2}=\frac{1}{4}\)

C


Why circles cannot be concentric? To my understanding the max distance the frog can go is 3 mtrs from its starting point, so this gives our larger circle of radius 3. the questions asks the region within the 1mtr radius circle. so favorable outcome is area of smaller circle and total outcomes is area of large circle. But, obviously i am making some mistake. Please help in identifying it. thanks chetan2u , Bunuel
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Re: A frog makes 3 jumps, each exactly 1 meter long. The directions of the [#permalink] New post   18 Oct 2020, 23:27
Bunuel Please post explanation to this question.
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Re: A frog makes 3 jumps, each exactly 1 meter long. The directions of the [#permalink] New post   20 Oct 2020, 01:09
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Given

    • A frog makes 3 jumps, each exactly 1 meter long.
    • The directions of the jumps are chosen independently at random.

To Find

    • The probability that the frog's final position is no more than 1 meter from its starting position.


Approach and Working Out

    • Visualization is key here. Please look at the diagram.

    o Here A is the starting position.
      o B is the position after the first jump and hence it is one meter away.
         B can be any point on the circle in amber but I have taken a specific one as it will not matter initially.
      o C is a point 1 meter from B so it can be any point on the circle colored in green.
         Here, we don’t know which point is C as it can even be at A or anyone along the circumference of the circle in green.
        o Now for the third jump, the distance between C can and D must be 1 m.
       Hence D will depend on C but since it is 1 m away so it must form a concentric circle around the point B. (As an upper limit)
        o So now the frog can be at any point within the circle in blue.
        o We need to find the probability of it being in amber.
        o So we need to find out the ratio of the area of the circle in amber to that of the circle in blue.
       That will be ¼ as the ratio of the radius is 1: 2.


Correct Answer: Option C
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Re: A frog makes 3 jumps, each exactly 1 meter long. The directions of the [#permalink] New post   20 Oct 2020, 21:03
EgmatQuantExpert wrote:
Given

    • A frog makes 3 jumps, each exactly 1 meter long.
    • The directions of the jumps are chosen independently at random.

To Find

    • The probability that the frog's final position is no more than 1 meter from its starting position.


Approach and Working Out

    • Visualization is key here. Please look at the diagram.

    o Here A is the starting position.
      o B is the position after the first jump and hence it is one meter away.
         B can be any point on the circle in amber but I have taken a specific one as it will not matter initially.
      o C is a point 1 meter from B so it can be any point on the circle colored in green.
         Here, we don’t know which point is C as it can even be at A or anyone along the circumference of the circle in green.
        o Now for the third jump, the distance between C can and D must be 1 m.
       Hence D will depend on C but since it is 1 m away so it must form a concentric circle around the point B. (As an upper limit)
        o So now the frog can be at any point within the circle in blue.
        o We need to find the probability of it being in amber.
        o So we need to find out the ratio of the area of the circle in amber to that of the circle in blue.
       That will be ¼ as the ratio of the radius is 1: 2.


Correct Answer: Option C


Kindly explain the fault in my logic:

I am assuming all the 3 circles in your diagram to be concentric. So as per that logic, the outermost circle will have a radius of 3units and ratio will be 1:9.
I am unable to understand that why the outermost circle cannot have 3units radius. Thank you.

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A frog makes 3 jumps, each exactly 1 meter long. The directions of the [#permalink] New post   20 Oct 2020, 23:41
Expert Reply
Capakki wrote:
EgmatQuantExpert wrote:
Given

    • A frog makes 3 jumps, each exactly 1 meter long.
    • The directions of the jumps are chosen independently at random.

To Find

    • The probability that the frog's final position is no more than 1 meter from its starting position.


Approach and Working Out

    • Visualization is key here. Please look at the diagram.

    o Here A is the starting position.
      o B is the position after the first jump and hence it is one meter away.
         B can be any point on the circle in amber but I have taken a specific one as it will not matter initially.
      o C is a point 1 meter from B so it can be any point on the circle colored in green.
         Here, we don’t know which point is C as it can even be at A or anyone along the circumference of the circle in green.
        o Now for the third jump, the distance between C can and D must be 1 m.
       Hence D will depend on C but since it is 1 m away so it must form a concentric circle around the point B. (As an upper limit)
        o So now the frog can be at any point within the circle in blue.
        o We need to find the probability of it being in amber.
        o So we need to find out the ratio of the area of the circle in amber to that of the circle in blue.
       That will be ¼ as the ratio of the radius is 1: 2.


Correct Answer: Option C


Kindly explain the fault in my logic:

I am assuming all the 3 circles in your diagram to be concentric. So as per that logic, the outermost circle will have a radius of 3units and ratio will be 1:9.
I am unable to understand that why the outermost circle cannot have 3units radius. Thank you.

Posted from my mobile device



    • All three cannot be concentric. As you can see that the first two circles cut each other.
    • Only the second circle in green and the third one in blue are concentric.
    • Most importantly it is very much important to understand that the first two circles cannot be concentric. Once you understand how the first two circles exist on a plane it will be easier to comprehend it.
      • When you plot B, you can just take any point at random as all the points will be identical. But for C and D we will have multiple possibilities which will give rise to the concentric circles.
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Re: A frog makes 3 jumps, each exactly 1 meter long. The directions of the [#permalink] New post   20 Oct 2020, 23:51
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Since frog is jumping one meter long ,there are four directions in which it can jumpl,so we can select 3 instance from 4c3 and jump is one instance ,so probablity should be 1/4c3
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Re: A frog makes 3 jumps, each exactly 1 meter long. The directions of the [#permalink] New post   01 Dec 2023, 07:32
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Re: A frog makes 3 jumps, each exactly 1 meter long. The directions of the [#permalink]
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