We cut [0,1] interval into 6 pieces by using uniformly chosen 5 separate points. Then, what is the probability that the lengths of all pieces are less than half?
I approach this problem by a standard probability approach. But it is 6 dimensional so I am hard to imagine the shape of the associated probability integration.
First draw 5 uniform points and label them as 1,2,3,4,5 from left to right. Then,
The desired probability is
$$ 5! * P(X_1<1/2, X_2-X_1< 1/2, X_3-X_2<1/2, X_4-X_3<1/2, X_5-X_4<1/2, 1-X_5<1/2) $$ where 5! is the possible order of 5 different points.
Hint: Maybe the following alternate problem is simpler (and helpful): What is the probability that, after choosing 6 points uniformly on a circle, one of the arcs is $\ge 180^\circ$?