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 <strike>a random</strike> *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 that *every* such geodesic is closed?

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![ZollDiscr][1]
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<sub>(Zoll Surface: Image from [Polthier&Schmies][2] via [this MO question][3])</sub>


  [1]: https://i.sstatic.net/rCTNz.gif
  [2]: http://www-sfb288.math.tu-berlin.de/Research/GEODESICS/Geodesic.html
  [3]: http://mathoverflow.net/a/150564/6094