Question

A point is chosen at random inside a circle. Find the probability p that the point is closer to the center of
the circle than to its circumfence
​

Answer 1

Step-by-step explanation:

Let R be radius of the circle

Distance of midpoint is  r=R2  

Now Area of bigger circle,  A=πR2  

Area of circle(area closer to centre) with r as radius  a=πr2=πR24  

Area of region closer to circumference  ac=A−a=πR2−πR24=3πR24  

Now, required probability

P=aac=πR243πR24  

Thus ,  

P=13  

Hope it helps!!

Also

Let's consider the following case :-

Consider the above disc to be a Compact disc of 800MB total space= area of big circle

Now since B is mid point and for even I have same approach as yours, area of small circle =200MB which is 1/4 th of total area

So remaining area is = 600MB

Now say the disc had 800 photos of 1MB each

And let's say the sectors in outer region of inner circle are damaged beyond recovery

Suppose the original disc had one photo of yours amongst the 800 photos

What the requirement as of now is the probability that your photo is available in the safe region.??

It's not 200 against 800

it's 200 against 600

P=200600=13  

However If the disc is a bulz eye dart game then the probability of hitting it withing the inner circle is  14  but the point closeness with centre against the circumference is  13  

Hope it helps!!Every person has answered 1/4

Basis the probability of favourable outcomes against all odds (favourable+unfavorable)

This works fine for position the point within the smalar area against the entire area

However question specifies the fight is for jurisdiction of centre against circumference

Which implies the conditional probability is for

Inner area against other area

Favourable odds against unfavorable odds

1/4 against 3/4

Hence 1/3