## Eyad 3 years ago Please I need some intelligence,Patience...To help me in solving this question.Ty.

An ant stands in the middle of a circle (3 metres in diameter) and walks in a straight line at a random angle from 0 to 360 degrees. Problem is, it can only walk one metre before it needs a break. The ant has the memory of a fish and forgets what direction it has just walked in.. Anyway, after the break, it gets all dizzy and thus chooses another random direction from 0 to 360 in an attempt to escape the circle again. As you can well imagine, it could escape the circle after just 2 walks (just one break needed). Or... it could take 20,000 walks (19,999 breaks needed)!! There might even be the very slim possibility it might take 20,000^20,000 walks.... What is the average amount of walks required for the ant to escape the circle?

@satellite73 ,@Calcmathlete

3. satellite73

there is no snap solution to this problem i think try googling it and you will come up with many many posts, since apparently this is a common problem no good solutions though

4. TuringTest

am I on the completely wrong track to say that since he has a remaining 0.5 to go to escape the circle that $$\cos\theta\ge\frac12\implies-\frac\pi6\le\theta\le\frac\pi6$$

5. amrit110

not completely wrong....ur on the right track...:)

6. TuringTest

I mean $-\frac\pi3\le\theta\le\frac\pi3$

7. amrit110

yes...now u just compute the average

8. amrit110

u do realize the ant neednt exactly go by ur condition!!

9. TuringTest

so that is 1/3 of a circle... so there is a 1/3 chance he gets out on move 2 where do I include the average, I still don't quite get it :/

10. amrit110

true...good enough. avearge is when u consider a particular no. of test cases. so if u consider 100, there is 33% chance of getting out on the 2nd move. now all u have to do is calculate the remaining chances for the other angles he might take....now although this is quite random, the probability for each angle range decreases n goes like a geometric series

11. TuringTest

yeah that's where the problem for me is... a geometric series is a good idea if we can set one up, but since the exact angle that he returns on is variable that won't be easy

12. TuringTest

like he could go|dw:1341369002782:dw|(move 1)

13. amrit110

yup...its a tedious problem...but i think a general solution is difficult but not impossible to obtain.

14. TuringTest

|dw:1341369028085:dw|that leaves him with a 0% chance of leaving the circle in the next move, or|dw:1341369061211:dw|in which case he has some other odds of leaving the circle in the next move

15. TuringTest

@Zarkon probability problem, interested?

16. amrit110

for all u know, there is even the possiblity like the question itself that the ant might not come out at all if he keeps taking angles that put him back inside.

17. TuringTest

that was suggested by @Outkast3r09 on an earlier post, but @Eyad says there is a definite answer, so...

18. amrit110

19. TuringTest

well that still is not as elegant as I was hoping where's the actual answer now...?

20. TuringTest

11.4 is the "expected value" so I guess that's the answer