DAY 86

Comment

One approach is to consider a more extreme version of the problem:
imagine that the angle of sector Y is very, very small, so that the chance of landing on Y is quite remote; then the chance of landing on Y and then not on Y will be (slightly) more remote still.

Answer

X then not X is more likely than Y then not Y.

Readers' methods

Betty Mulder, Seattle:
The probability of landing on Y looks to be about 1 in 16;
so the probability of Y then not Y is even less than this (1 in 17, say).
On the other hand, the probability of landing on X is one half, and so is the probability of not landing on X;
a half then a half is a quarter, which is much more than 1 in 17.

Margaret T, 17
x = angle in sector X (=180), y = angle in sector Y;
p(X then not X) = x/360 * x/360 = x^2/129600;
x = 180, therefore p(X) = 32400/129600.
p(Y then not Y) = y/360 * (360 - y)/360 = y(360 - y)/129600;
if p(X then not X) < p(Y then not Y) then 32400 < 360y - y^2,
so y^2 - 360y + 32400 < 0,
so (y - 180)^2 < 0;
but this is not possible, as (y - 180)^2 will always be positive [or zero, when y = 180],
so p(X then not X) must be more likely than p(Y then not Y)
[unless y = 180, when they are equally likely].