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

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Math Expert
Joined: 02 Sep 2009
Posts: 58428
A frog makes 3 jumps, each exactly 1 meter long. The directions of the  [#permalink]

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31 Mar 2019, 21:13
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Difficulty:

75% (hard)

Question Stats:

31% (02:13) correct 69% (02:13) wrong based on 16 sessions

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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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Math Expert
Joined: 02 Aug 2009
Posts: 8006
A frog makes 3 jumps, each exactly 1 meter long. The directions of the  [#permalink]

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31 Mar 2019, 21:52
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
Attachments

frog.png [ 31.65 KiB | Viewed 243 times ]

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A frog makes 3 jumps, each exactly 1 meter long. The directions of the   [#permalink] 31 Mar 2019, 21:52
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