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Melody

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Melody  11 févr. 2022
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+1

Earlier errors are now fixed.  I hope there are no more.

 

Question: Two points on a circle of radius 1 are chosen at random. Find the probability that the distance between the two points is at most 1.

 

I have spent a long time on this, I think this answer is correct.

 

Orient your  circle so that the interval joining the 2 points that you choose is above and parallel to the diameter.

The green semicircle is the top of the original circle.

If I draw  1 unit intervals paralel to the diameter how much of semicircle will not be included.

 

I needed a curve 1 unit to right of the original and the area in the bricked in area indicatesw the left over.

 

So the prob of the points being 1 unit or less apart is the  clear green area over the whole green semicircle.

 

Area of Simicircle = π2

 

Area of the bricked sector sector =

 60360π=π6

 

Area of bricked segment

=π6121120.52=π61232=π634=2π3312

 

Total bricked area 

=2π12+2π123312=4π3312

 

Area on non-bricked section

 

 

6π124π3312=2π+3312

 

 

Probability that the points will be less than or equal to 1 unit apart 

=2π+3312÷π2=33+2π6π61%

 

 

Hopefully I have now fixed all the errors.

 

 

 

 

 

 

23 avr. 2023