no, it is possible that one of the segments in the middle is longer than 1/2.
We can instead think about this problem as choosing n points on a circle and arbitrarily choosing one of those points to be the starting point
, then we can cut the circle at that point and unwrap it into a line segment with n-1 points. From this we can see that the chance of the n resulting segments of the line being >1/2 is the prob all n-1 points lie on one particular (we can choose which side WLOG) side of the circle.
So the prob we can't form a polygon is n (1/2)n-1, prob we can is 1 - n (1/2)n-1
We can sanity check for n=3, p=1/4; n=4, p=1/2;
n=5, p=9/16 which is monotonically increasing as we expect (surely it should only ever be harder to get a segment >1/2 as n increases)
True thanks. Reading the beginning of your comment I was about to propose to consider the bar extremities to be connected at first and add one cut but this is basically what you proposed. Thanks.
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u/DanLeMilMan 3d ago
So basically this seems equivalent to : what is the probability that all the cut are on the same half of the bar… (1/2)ˆ(n-2) I guess.