P6.js

let mathjs = require("mathjs");

function getRandomPointOnSphere() {
	let o = 2 * Math.PI * Math.random();
	let a = Math.acos(2 * Math.random() - 1);
	return { o, a };
}

function polarToCarth(o) {
	let x = Math.sin(o.o) * Math.cos(o.a);
	let y = Math.sin(o.o) * Math.sin(o.a);
	let z = Math.cos(o.o);
	return [x, y, z];
}

function run() {
	let [x1, y1, z1] = polarToCarth(getRandomPointOnSphere());
	let [x2, y2, z2] = polarToCarth(getRandomPointOnSphere());
	let [x3, y3, z3] = polarToCarth(getRandomPointOnSphere());
	let [x4, y4, z4] = polarToCarth(getRandomPointOnSphere());

	let x = 0;
	let y = 0;
	let z = 0;

	let d0 = mathjs.det([
		[x1, y1, z1, 1],
		[x2, y2, z2, 1],
		[x3, y3, z3, 1],
		[x4, y4, z4, 1]
	]);

	let d1 = mathjs.det([
		[x, y, z, 1],
		[x2, y2, z2, 1],
		[x3, y3, z3, 1],
		[x4, y4, z4, 1]
	]);

	let d2 = mathjs.det([
		[x1, y1, z1, 1],
		[x, y, z, 1],
		[x3, y3, z3, 1],
		[x4, y4, z4, 1]
	]);

	let d3 = mathjs.det([
		[x1, y1, z1, 1],
		[x2, y2, z2, 1],
		[x, y, z, 1],
		[x4, y4, z4, 1]
	]);

	let d4 = mathjs.det([
		[x1, y1, z1, 1],
		[x2, y2, z2, 1],
		[x3, y3, z3, 1],
		[x, y, z, 1]
	]);

	if (
		(d0 >= 0 && d1 >= 0 && d2 >= 0 && d3 >= 0 && d4 >= 0) ||
		(d0 < 0 && d1 < 0 && d2 < 0 && d3 < 0 && d4 < 0)
	) {
		return true;
	} else {
		return false;
	}
}

let yes = 0;
let no = 0;
for (var i = 0; i < 1000000; i++) {
	if (i % 1000 == 0) {
		console.log(i);
	}
	if (run()) {
		yes++;
	} else {
		no++;
	}
}
console.log(yes, no, yes / 1000000);