At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
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?
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
am I on the completely wrong track to say that since he has a remaining 0.5 to go to escape the circle that \(\cos\theta\ge\frac12\implies-\frac\pi6\le\theta\le\frac\pi6\)
not completely wrong....ur on the right track...:)
I mean \[-\frac\pi3\le\theta\le\frac\pi3\]
yes...now u just compute the average
u do realize the ant neednt exactly go by ur condition!!
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 :/
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
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
like he could go|dw:1341369002782:dw|(move 1)
yup...its a tedious problem...but i think a general solution is difficult but not impossible to obtain.
|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
@Zarkon probability problem, interested?
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.
that was suggested by @Outkast3r09 on an earlier post, but @Eyad says there is a definite answer, so...
http://math.stackexchange.com/questions/133851/ant-in-a-circle. this page has a good solution.
well that still is not as elegant as I was hoping where's the actual answer now...?
11.4 is the "expected value" so I guess that's the answer
Ok Then Ty everyone :) Ty @TuringTest ,@amrit110 :ty for the site its useful :)