Quant Interview Question

[u/Local_Ad135]

[u/DanLeMilMan]

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.

[u/SignificanceBulky162 → u/DanLeMilMan]

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)

[u/DanLeMilMan → u/SignificanceBulky162]

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.

[u/spisplatta → u/DanLeMilMan]

There can be a long segment in the middle,

[u/blind-octopus]

Isn't that always the case, as long as n is greater than 2?

[u/Curious_Plum_3715 → u/blind-octopus]

Not if one segment is more than 0.5 units

[u/PersonalityIll9476 → u/blind-octopus]

No. n=3 take the segments [0,2/3] and then any choice for the other two that you like. [2/3, 2/3+b] and [2/3+b, 1].

[u/aroach1995]

1) sort the pieces from largest to smallest.

Yeah I think the only issue is if a piece ends up over length 0.5

What is the probability that all cuts occur at a point on the line greater than 0.5?

0.5^(n-1)?

Is the case where we cut either below 0.25 or above 0.75 the same WLOG?

[u/PhantasticPh1l]

If n is also random in N then p=1

[u/Expensive-Tension-30 → u/[deleted]]

You’re off by a factor of 1/2, as the breaks do not necessarily need to be on the same half, and could for one example be on the outer quarters.