Let $S$ be a closed surface embedded in $\mathbb{R}^3$,
let's say of genus zero.
I seek examples of $S$ with the following property:
If one selects ~~a random~~ *any* point $p$ on $S$, and a random
direction $u$ tangent to $S$ at $p$, then the geodesic
issuing from $p$ in direction $u$ is a closed geodesic with
positive probability.
[Question modified to reflect @alvarezpaiva's comment.]

I know this is trivially true for Zoll surfaces,
on which *every* geodesic is closed; see figure below.
But are there non-Zoll $S$ where geodesics are prevalent
enough to yield a positive probability (perhaps $1$)?
Or, to be more explicit:

Q. If, for every $p \in S$, closed geodesics issuing from $p$ occur with probablility $1$ among all tangent directions $u$, does that imply thateverysuch geodesic is closed?

_{(Zoll Surface: Image from Polthier&Schmies via this MO question)}

everypoint $p$ there is a positive probability that a geodesic through $p$ be closed? – alvarezpaiva Feb 20 at 12:44notbe the same problem. I think that if you take a metric of revolution on the sphere the initial conditions for closed geodesics may be dense in the unit tangent bundle (the invariant tori with rational rotation numbers are foliated by periodic orbits). – alvarezpaiva Feb 21 at 21:55