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Bunuel Please post explanation to this question.
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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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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
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Capakki
EgmatQuantExpert
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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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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