About “they lie on a perfectly straight line is zero”，here is a slightly overengineered proof.
Write the three points as
\( P_i=(X_i,Y_i) \)
The sample space is
\[ D^3\subset\mathbb R^6 \]
where \(D\) is the unit disk.
The three points are collinear if and only if
\[\begin{vmatrix} X_1&Y_1&1\\ X_2&Y_2&1\\ X_3&Y_3&1 \end{vmatrix}=0 \]
In other words,
\[ f(X_1,Y_1,X_2,Y_2,X_3,Y_3)=0 \]
where \(f\) is a polynomial in six variables.
Thus the set of collinear triples is
\(A=f^{-1}(0) \)
an algebraic hypersurface in \(\mathbb R^6\). Its dimension is at most \(5\), and hence it has \(6\)-dimensional Lebesgue measure zero.
On the other hand, the joint distribution of the three independently and uniformly chosen points is absolutely continuous with respect to \(6\)-dimensional Lebesgue measure.
Therefore,
P(A)=0
majer
npub1wc…xv88t
2026-10-06 06:56:05 UTC
in reply to nevent1q…pmmc
Author Public Key
npub1wckh0566nx8y9myxmmt3l0292prhwgtsk8cghh2fdq49h3sknwksjxv88tSeen on
wss://relay.momostr.pinkPublished at
2026-10-06 06:56:05 UTCEvent JSON
{
"id": "a5fa36da5eeab7c9341954a2f1c3035ff1fbb1856c74a122cc79ca28109d9b6a",
"pubkey": "762d77d35a998e42ec86ded71fbd455047772170b1f08bdd49682a5bc6169bad",
"created_at": 1791269765,
"kind": 1,
"tags": [
[
"e",
"e3d6ad2fb56d949e027d33da1a4d1177f2c3286d645e3811876dc313e11d24be",
"",
"root",
"563821004c3e4e4f4fbef74a2657db3299a0e71a78214bf97db0b6ca05fb73d3"
],
[
"p",
"563821004c3e4e4f4fbef74a2657db3299a0e71a78214bf97db0b6ca05fb73d3"
],
[
"proxy",
"https://mathstodon.xyz/@majer/117392655374817060",
"web"
],
[
"proxy",
"https://mathstodon.xyz/users/majer/statuses/117392655374817060",
"activitypub"
],
[
"L",
"pink.momostr"
],
[
"l",
"pink.momostr.activitypub:https://mathstodon.xyz/users/majer/statuses/117392655374817060",
"pink.momostr"
],
[
"-"
]
],
"content": "About “they lie on a perfectly straight line is zero”，here is a slightly overengineered proof.\n\nWrite the three points as\n\n\\( P_i=(X_i,Y_i) \\) \n\nThe sample space is\n\\[ D^3\\subset\\mathbb R^6 \\] \n\nwhere \\(D\\) is the unit disk.\n\nThe three points are collinear if and only if\n\\[\\begin{vmatrix} X_1\u0026Y_1\u00261\\\\ X_2\u0026Y_2\u00261\\\\ X_3\u0026Y_3\u00261 \\end{vmatrix}=0 \\] \n\nIn other words,\n\\[ f(X_1,Y_1,X_2,Y_2,X_3,Y_3)=0 \\] \n\nwhere \\(f\\) is a polynomial in six variables.\n\nThus the set of collinear triples is\n\\(A=f^{-1}(0) \\) \n\nan algebraic hypersurface in \\(\\mathbb R^6\\). Its dimension is at most \\(5\\), and hence it has \\(6\\)-dimensional Lebesgue measure zero.\n\nOn the other hand, the joint distribution of the three independently and uniformly chosen points is absolutely continuous with respect to \\(6\\)-dimensional Lebesgue measure.\n\nTherefore,\n\nP(A)=0\n\t",
"sig": "b6edce267caf0d601128f0cada82991c30b9d3a01ce6b4fe407add6f33413765725ef0309b031f6dda156b1f9b19bb8d70e002733157b897d04d30ca19ed76f9"
}