18/06/2026
Probability of Ball Hitting Post/Crossbar vs Goal
1. Assumptions (Model Setup)
Consider a football shot taken from distance ( D ) toward a goal.
* Goal width: ( W )
* Ball radius: ( r_b )
* Post radius: ( r_p )
* Effective collision radius:
[
R = r_b + r_p
]
We assume:
* Shot direction is described by angle ( \theta )
* Small angular error around target direction
* No goalkeeper effect
2. Geometry of the Goal
At distance ( D ), horizontal position of ball impact:
[
y = D \tan\theta
]
Goal boundaries (adjusted for ball radius):
* Left boundary: ( -\frac{W}{2} + R )
* Right boundary: ( \frac{W}{2} - R )
---
3. Classification Conditions
(i) Goal
[
-\frac{W}{2} + R \le y \le \frac{W}{2} - R
]
(ii) Post / Crossbar hit
Occurs when ball is within collision band:
[
\frac{W}{2} - R < |y| \le \frac{W}{2} + R
]
---
(iii) Miss
[
|y| > \frac{W}{2} + R
]
---
4. Statistical Model of Shooting
Assume shooting angle follows normal distribution:
[
\theta \sim \mathcal{N}(0, \sigma^2)
]
For small angles:
[
y \approx D\theta
]
Thus:
[
y \sim \mathcal{N}(0, D^2\sigma^2)
]
Let:
[
\sigma_y = D\sigma
]
---
5. Probability Formulation
We convert geometry into probability intervals.
# # # Goal probability:
[
P_{goal} = P\left(-a \le y \le a\right)
]
where:
[
a = \frac{W}{2} - R
]
---
# # # Post probability:
[
P_{post} = P\left(a < |y| \le b\right)
]
where:
[
b = \frac{W}{2} + R
]
---
6. Using Normal Distribution
Let ( \Phi ) be the standard normal CDF:
[
Z = \frac{y}{\sigma_y}
]
Then:
Goal:
[
P_{goal} = \Phi\left(\frac{a}{\sigma_y}\right) - \Phi\left(-\frac{a}{\sigma_y}\right)
]
---
Post:
[
P_{post} =
\left[\Phi\left(\frac{b}{\sigma_y}\right) - \Phi\left(\frac{a}{\sigma_y}\right)\right]
+
\left[\Phi\left(-\frac{a}{\sigma_y}\right) - \Phi\left(-\frac{b}{\sigma_y}\right)\right]
]
---
Miss:
[
P_{miss} = 1 - (P_{goal} + P_{post})
]
---
7. Interpretation
This model shows:
* Goal probability depends mainly on **variance of shooting accuracy ( \sigma )**
* Post/crossbar hits occur in a **thin boundary region**
* Increasing distance ( D ) increases spread ( \sigma_y ), raising miss and post probability
---
8. Key Insight
Even though the post is physically small, the probability is not purely geometric:
> It is dominated by the overlap between a Gaussian error distribution and a narrow boundary region around the goal frame.
---
9. Final Form (Compact)
[
y \sim \mathcal{N}(0, D^2\sigma^2)
]
[
P_{goal}, P_{post}, P_{miss}
\quad \text{from interval integration over } y
]
