Geometry of a Fluke

[Jim Leff]

Very brief and entertaining video clip

It takes a couple viewings to see exactly what's going on here: the ball flew straight into the pitcher's glove (and he then had to manage his body’s reaction).

I'm curious about the odds of a ball reaching a 11cm x 11cm target from a distance of 18m. Maybe assume, for simplicity, that it's equally likely to be hit anywhere within a 90° horizontal angle (let's discount foul balls), and at any vertical angle of 0 - 45°. Pitcher's mound is about 18m from home plate.

Can anyone help?

[neufer]

Very brief and entertaining video clip

It takes a couple viewings to see exactly what's going on here: the ball flew straight into the pitcher's glove (and he then had to manage his body’s reaction).

I'm curious about the odds of a ball reaching a 11cm x 11cm target from a distance of 18m. Maybe assume, for simplicity, that it's equally likely to be hit anywhere within a 90° horizontal angle (let's discount foul balls), and at any vertical angle of 0 - 45°. Pitcher's mound is about 18m from home plate.

Can anyone help?

To first order (for a ball traveling that fast) it must be something like 2 in (1800 cm/11 cm)².

[Jim Leff]

Thanks! 163.6 squared is 26765. So the odds are something like 2 in 26765, or 1 in 13382?

[bystander]

Did he throw out the guy on second? He didn't tag up.

[Jim Leff]

Good point. Not sure.

[neufer]

Thanks! 163.6 squared is 26765. So the odds are something like 2 in 26765, or 1 in 13382?

It is an absolute fact that the probability of

• 1) hitting a 22 cm x 22 cm square in the vicinity of the pitcher is 4 times the
  probability of hitting a 11 cm x 11 cm square in the vicinity of the pitcher
  
  and that the probability of
  
  2) hitting a 33 cm x 33 cm square in the vicinity of the pitcher is 9 times the
  probability of hitting a 11 cm x 11 cm square in the vicinity of the pitcher

The only question is at what point within a 18 meter radius shell surrounding the batter
does the probability of hitting any given 11 cm x 11 cm square drop off from that of
the stated perfect back-scatter 11 cm x 11 cm square situation.

[Jim Leff]

If I understand your question correctly, it was addressed when I suggested we assume, for simplicity, that it's equally likely to be hit anywhere within a 90° horizontal angle and at any vertical angle of 0 - 45°.

However I’m still not sure if I’ve translated your previous reply into the correct “1 in xx” statement of odds.

[neufer]

If I understand your question correctly, it was addressed when I suggested we assume, for simplicity, that it's equally likely to be hit anywhere within a 90° horizontal angle and at any vertical angle of 0 - 45°.

However I’m still not sure if I’ve translated your previous reply into the correct “1 in xx” statement of odds.

I was actually just trying to explain my logic in response
to your immediate reply (which you have since deleted).

If, in fact,
"it's equally likely to be hit anywhere within
a 90° horizontal angle and at any vertical angle of 0 - 45°" = 1.11 steradians

as compared to the target area of (11 cm/1800 cm)² steradians,

then the answer (for a very fast ball) is ~1 in 29,750.

One should consider pop up flies & ground balls as other non-foul playable hits
but ~1 in 29,750 is probably in the right ballpark.

[Jim Leff]

I haven’t deleted anything.

Thanks.