This market resolves to the value of p(heads) (the probability of flipping heads) for a biased coin. The probability will be multiplied by 100 to express it as a percentage.
The first 3 flips are shown. I will flip the coin again once per day and update the results. If I forget to update the description, please remind me to do so.
Anyone who has bet at least 1000 mana on this question can DM me or publicly comment asking for the market to be resolved.
p(heads) was sampled uniformly randomly between 0 and 1. The salted sha256 hash of p(heads) is: 5c0d2da2236bcccb8c47653ec4860a113266e65d95c9d341b4eea720e5b06525.
When the market resolves, I will publish:
- The plaintext message used to generate the hash
- The Python code used to generate the model so the results can be independently verified
Start: 2024-08-25
Timestamp: 2025-01-04
Elapsed: 132
Samples: 587
Flip Date Res p_hat
1 2024-08-25 T 0.00
2 2024-08-25 H 0.50
3 2024-08-25 H 0.67
4 2024-08-26 T 0.50
5 2024-08-27 H 0.60
6 2024-08-28 T 0.50
7 2024-08-29 T 0.43
8 2024-08-30 H 0.50
9 2024-08-31 T 0.44
10 2024-09-01 T 0.40
11 2024-09-02 H 0.45
12 2024-09-03 T 0.42
13 2024-09-04 T 0.38
14 2024-09-05 H 0.43
15 2024-09-06 H 0.47
16 2024-09-07 T 0.44
17 2024-09-08 H 0.47
18 2024-09-09 H 0.50
19 2024-09-10 H 0.53
20 2024-09-11 H 0.55
21 2024-09-12 T 0.52
22 2024-09-13 T 0.50
23 2024-09-14 T 0.48
24 2024-09-14 H 0.50
25 2024-09-14 T 0.48
26 2024-09-14 T 0.46
27 2024-09-14 T 0.44
28 2024-09-15 T 0.43
29 2024-09-15 T 0.41
30 2024-09-15 H 0.43
31 2024-09-15 T 0.42
32 2024-09-15 H 0.44
33 2024-09-16 H 0.45
34 2024-09-16 T 0.44
35 2024-09-16 H 0.46
36 2024-09-16 H 0.47
37 2024-09-16 H 0.49
38 2024-09-17 H 0.50
39 2024-09-17 H 0.51
40 2024-09-17 H 0.53
41 2024-09-17 T 0.51
42 2024-09-17 T 0.50
43 2024-09-18 H 0.51
44 2024-09-18 T 0.50
45 2024-09-18 T 0.49
46 2024-09-18 T 0.48
47 2024-09-18 H 0.49
48 2024-09-19 T 0.48
49 2024-09-19 H 0.49
50 2024-09-19 H 0.50
51 2024-09-19 H 0.51
52 2024-09-19 T 0.50
53 2024-09-20 H 0.51
54 2024-09-20 T 0.50
55 2024-09-20 H 0.51
56 2024-09-20 H 0.52
57 2024-09-20 H 0.53
58 2024-09-21 T 0.52
59 2024-09-21 T 0.51
60 2024-09-21 H 0.52
61 2024-09-21 H 0.52
62 2024-09-21 H 0.53
63 2024-09-22 H 0.54
64 2024-09-22 H 0.55
65 2024-09-22 T 0.54
66 2024-09-22 T 0.53
67 2024-09-22 H 0.54
68 2024-09-23 T 0.53
69 2024-09-23 H 0.54
70 2024-09-23 H 0.54
71 2024-09-23 H 0.55
72 2024-09-23 T 0.54
73 2024-09-24 H 0.55
74 2024-09-24 H 0.55
75 2024-09-24 T 0.55
76 2024-09-24 T 0.54
77 2024-09-24 T 0.53
78 2024-09-25 T 0.53
79 2024-09-25 H 0.53
80 2024-09-25 H 0.54
81 2024-09-25 H 0.54
82 2024-09-25 T 0.54
83 2024-09-26 T 0.53
84 2024-09-26 T 0.52
85 2024-09-26 T 0.52
86 2024-09-26 T 0.51
87 2024-09-26 T 0.51
88 2024-09-27 T 0.50
89 2024-09-27 H 0.51
90 2024-09-27 T 0.50
91 2024-09-27 H 0.51
92 2024-09-27 T 0.50
93 2024-09-28 H 0.51
94 2024-09-28 H 0.51
95 2024-09-28 T 0.51
96 2024-09-28 T 0.50
97 2024-09-28 T 0.49
98 2024-09-29 H 0.50
99 2024-09-29 H 0.51
100 2024-09-29 T 0.50
101 2024-09-29 T 0.50
102 2024-09-29 T 0.49
103 2024-09-30 T 0.49
104 2024-09-30 H 0.49
105 2024-09-30 T 0.49
106 2024-09-30 H 0.49
107 2024-09-30 T 0.49
108 2024-10-01 H 0.49
109 2024-10-01 H 0.50
110 2024-10-01 T 0.49
111 2024-10-01 T 0.49
112 2024-10-01 T 0.48
113 2024-10-02 T 0.48
114 2024-10-02 T 0.47
115 2024-10-02 H 0.48
116 2024-10-02 T 0.47
117 2024-10-02 T 0.47
118 2024-10-03 T 0.47
119 2024-10-03 H 0.47
120 2024-10-03 T 0.47
121 2024-10-03 T 0.46
122 2024-10-03 T 0.46
123 2024-10-04 T 0.46
124 2024-10-04 H 0.46
125 2024-10-04 T 0.46
126 2024-10-04 T 0.45
127 2024-10-04 H 0.46
128 2024-10-05 T 0.45
129 2024-10-05 T 0.45
130 2024-10-05 H 0.45
131 2024-10-05 T 0.45
132 2024-10-05 T 0.45
133 2024-10-06 H 0.45
134 2024-10-06 T 0.45
135 2024-10-06 T 0.44
136 2024-10-06 T 0.44
137 2024-10-06 H 0.45
138 2024-10-07 T 0.44
139 2024-10-07 H 0.45
140 2024-10-07 H 0.45
141 2024-10-07 T 0.45
142 2024-10-07 H 0.45
143 2024-10-08 T 0.45
144 2024-10-08 T 0.44
145 2024-10-08 T 0.44
146 2024-10-08 H 0.45
147 2024-10-08 H 0.45
148 2024-10-09 H 0.45
149 2024-10-09 H 0.46
150 2024-10-09 T 0.45
151 2024-10-09 T 0.45
152 2024-10-09 H 0.45
153 2024-10-10 T 0.45
154 2024-10-10 T 0.45
155 2024-10-10 T 0.45
156 2024-10-10 H 0.45
157 2024-10-10 H 0.45
158 2024-10-11 T 0.45
159 2024-10-11 H 0.45
160 2024-10-11 T 0.45
161 2024-10-11 H 0.45
162 2024-10-11 T 0.45
163 2024-10-12 H 0.45
164 2024-10-12 T 0.45
165 2024-10-12 T 0.45
166 2024-10-12 T 0.45
167 2024-10-12 H 0.45
168 2024-10-13 H 0.45
169 2024-10-13 T 0.45
170 2024-10-13 H 0.45
171 2024-10-13 T 0.45
172 2024-10-13 H 0.45
173 2024-10-14 H 0.46
174 2024-10-14 T 0.45
175 2024-10-14 T 0.45
176 2024-10-14 T 0.45
177 2024-10-14 T 0.45
178 2024-10-15 H 0.45
179 2024-10-15 H 0.45
180 2024-10-15 T 0.45
181 2024-10-15 T 0.45
182 2024-10-15 H 0.45
183 2024-10-16 H 0.45
184 2024-10-16 T 0.45
185 2024-10-16 H 0.45
186 2024-10-16 T 0.45
187 2024-10-16 T 0.45
188 2024-10-17 T 0.45
189 2024-10-17 H 0.45
190 2024-10-17 H 0.45
191 2024-10-17 T 0.45
192 2024-10-17 T 0.45
193 2024-10-18 T 0.45
194 2024-10-18 H 0.45
195 2024-10-18 H 0.45
196 2024-10-18 T 0.45
197 2024-10-18 T 0.45
198 2024-10-19 T 0.44
199 2024-10-19 H 0.45
200 2024-10-19 T 0.45
201 2024-10-19 T 0.44
202 2024-10-19 T 0.44
203 2024-10-20 T 0.44
204 2024-10-20 H 0.44
205 2024-10-20 H 0.44
206 2024-10-20 H 0.45
207 2024-10-20 T 0.44
208 2024-10-21 H 0.45
209 2024-10-21 T 0.44
210 2024-10-21 T 0.44
211 2024-10-21 T 0.44
212 2024-10-21 T 0.44
213 2024-10-22 T 0.44
214 2024-10-22 T 0.43
215 2024-10-22 T 0.43
216 2024-10-22 T 0.43
217 2024-10-22 H 0.43
218 2024-10-23 H 0.44
219 2024-10-23 H 0.44
220 2024-10-23 T 0.44
221 2024-10-23 T 0.43
222 2024-10-23 T 0.43
223 2024-10-24 H 0.43
224 2024-10-24 T 0.43
225 2024-10-24 H 0.44
226 2024-10-24 T 0.43
227 2024-10-24 T 0.43
228 2024-10-25 T 0.43
229 2024-10-25 T 0.43
230 2024-10-25 T 0.43
231 2024-10-25 T 0.42
232 2024-10-25 H 0.43
233 2024-10-26 T 0.42
234 2024-10-26 H 0.43
235 2024-10-26 T 0.43
236 2024-10-26 T 0.42
237 2024-10-26 T 0.42
238 2024-10-27 H 0.42
239 2024-10-27 T 0.42
240 2024-10-27 H 0.42
241 2024-10-27 H 0.43
242 2024-10-27 H 0.43
243 2024-10-28 H 0.43
244 2024-10-28 H 0.43
245 2024-10-28 T 0.43
246 2024-10-28 T 0.43
247 2024-10-28 T 0.43
248 2024-10-29 T 0.43
249 2024-10-29 H 0.43
250 2024-10-29 H 0.43
251 2024-10-29 T 0.43
252 2024-10-29 T 0.43
253 2024-10-30 T 0.43
254 2024-10-30 H 0.43
255 2024-10-30 H 0.43
256 2024-10-30 T 0.43
257 2024-10-30 T 0.43
258 2024-10-31 H 0.43
259 2024-10-31 H 0.43
260 2024-10-31 T 0.43
261 2024-10-31 H 0.43
262 2024-10-31 T 0.43
263 2024-11-01 T 0.43
264 2024-11-01 H 0.43
265 2024-11-01 H 0.43
266 2024-11-01 H 0.44
267 2024-11-01 T 0.43
268 2024-11-02 T 0.43
269 2024-11-02 T 0.43
270 2024-11-02 T 0.43
271 2024-11-02 H 0.43
272 2024-11-02 H 0.43
273 2024-11-03 T 0.43
274 2024-11-03 T 0.43
275 2024-11-03 H 0.43
276 2024-11-03 T 0.43
277 2024-11-03 T 0.43
278 2024-11-04 H 0.43
279 2024-11-04 H 0.43
280 2024-11-04 H 0.44
281 2024-11-04 T 0.43
282 2024-11-04 T 0.43
283 2024-11-05 T 0.43
284 2024-11-05 T 0.43
285 2024-11-05 H 0.43
286 2024-11-05 H 0.43
287 2024-11-05 T 0.43
288 2024-11-06 T 0.43
289 2024-11-06 H 0.43
290 2024-11-06 T 0.43
291 2024-11-06 T 0.43
292 2024-11-06 H 0.43
293 2024-11-07 T 0.43
294 2024-11-07 T 0.43
295 2024-11-07 H 0.43
296 2024-11-07 T 0.43
297 2024-11-07 H 0.43
298 2024-11-08 T 0.43
299 2024-11-08 T 0.43
300 2024-11-08 T 0.43
301 2024-11-08 T 0.43
302 2024-11-08 H 0.43
303 2024-11-09 T 0.43
304 2024-11-09 H 0.43
305 2024-11-09 H 0.43
306 2024-11-09 T 0.43
307 2024-11-09 T 0.43
308 2024-11-10 T 0.43
309 2024-11-10 H 0.43
310 2024-11-10 H 0.43
311 2024-11-10 T 0.43
312 2024-11-10 T 0.43
313 2024-11-11 H 0.43
314 2024-11-11 H 0.43
315 2024-11-11 H 0.43
316 2024-11-11 T 0.43
317 2024-11-11 T 0.43
318 2024-11-12 T 0.43
319 2024-11-12 T 0.43
320 2024-11-12 H 0.43
321 2024-11-12 T 0.43
322 2024-11-12 T 0.43
323 2024-11-13 T 0.42
324 2024-11-13 H 0.43
325 2024-11-13 H 0.43
326 2024-11-13 T 0.43
327 2024-11-13 H 0.43
328 2024-11-14 H 0.43
329 2024-11-14 T 0.43
330 2024-11-14 T 0.43
331 2024-11-14 T 0.43
332 2024-11-14 T 0.42
333 2024-11-15 H 0.43
334 2024-11-15 H 0.43
335 2024-11-15 T 0.43
336 2024-11-15 H 0.43
337 2024-11-15 T 0.43
338 2024-11-16 T 0.43
339 2024-11-16 T 0.42
340 2024-11-16 T 0.42
341 2024-11-16 H 0.43
342 2024-11-16 T 0.42
343 2024-11-17 H 0.43
344 2024-11-17 T 0.42
345 2024-11-17 T 0.42
346 2024-11-17 T 0.42
347 2024-11-17 H 0.42
348 2024-11-18 H 0.43
349 2024-11-18 T 0.42
350 2024-11-18 H 0.43
351 2024-11-18 T 0.42
352 2024-11-18 T 0.42
353 2024-11-19 T 0.42
354 2024-11-19 H 0.42
355 2024-11-19 H 0.43
356 2024-11-19 H 0.43
357 2024-11-19 T 0.43
358 2024-11-20 T 0.42
359 2024-11-20 H 0.43
360 2024-11-20 H 0.43
361 2024-11-20 H 0.43
362 2024-11-20 T 0.43
363 2024-11-21 H 0.43
364 2024-11-21 T 0.43
365 2024-11-21 T 0.43
366 2024-11-21 H 0.43
367 2024-11-21 T 0.43
368 2024-11-22 T 0.43
369 2024-11-22 T 0.43
370 2024-11-22 H 0.43
371 2024-11-22 H 0.43
372 2024-11-22 H 0.43
373 2024-11-23 H 0.43
374 2024-11-23 H 0.43
375 2024-11-23 T 0.43
376 2024-11-23 H 0.43
377 2024-11-23 T 0.43
378 2024-11-24 T 0.43
379 2024-11-24 T 0.43
380 2024-11-24 H 0.43
381 2024-11-24 H 0.43
382 2024-11-24 H 0.43
383 2024-11-25 T 0.43
384 2024-11-25 T 0.43
385 2024-11-25 T 0.43
386 2024-11-25 T 0.43
387 2024-11-25 H 0.43
388 2024-11-26 T 0.43
389 2024-11-26 H 0.43
390 2024-11-26 T 0.43
391 2024-11-26 T 0.43
392 2024-11-26 T 0.43
393 2024-11-27 H 0.43
394 2024-11-27 T 0.43
395 2024-11-27 T 0.43
396 2024-11-27 T 0.43
397 2024-11-27 H 0.43
398 2024-11-28 T 0.43
399 2024-11-28 H 0.43
400 2024-11-28 T 0.43
401 2024-11-28 T 0.43
402 2024-11-28 T 0.43
403 2024-11-29 H 0.43
404 2024-11-29 T 0.43
405 2024-11-29 T 0.42
406 2024-11-29 T 0.42
407 2024-11-29 H 0.43
408 2024-11-30 T 0.42
409 2024-11-30 T 0.42
410 2024-11-30 H 0.42
411 2024-11-30 H 0.43
412 2024-11-30 T 0.42
413 2024-12-01 T 0.42
414 2024-12-01 H 0.43
415 2024-12-01 H 0.43
416 2024-12-01 T 0.43
417 2024-12-01 T 0.42
418 2024-12-02 T 0.42
419 2024-12-02 T 0.42
420 2024-12-02 H 0.42
421 2024-12-02 H 0.43
422 2024-12-02 T 0.42
423 2024-12-03 H 0.43
424 2024-12-03 T 0.42
425 2024-12-03 T 0.42
426 2024-12-03 H 0.42
427 2024-12-03 T 0.42
428 2024-12-04 T 0.42
429 2024-12-04 T 0.42
430 2024-12-04 H 0.42
431 2024-12-04 T 0.42
432 2024-12-04 T 0.42
433 2024-12-05 T 0.42
434 2024-12-05 H 0.42
435 2024-12-05 H 0.42
436 2024-12-05 T 0.42
437 2024-12-05 T 0.42
438 2024-12-06 H 0.42
439 2024-12-06 H 0.42
440 2024-12-06 H 0.42
441 2024-12-06 H 0.43
442 2024-12-06 T 0.43
443 2024-12-07 H 0.43
444 2024-12-07 H 0.43
445 2024-12-07 H 0.43
446 2024-12-07 T 0.43
447 2024-12-07 H 0.43
448 2024-12-08 H 0.43
449 2024-12-08 T 0.43
450 2024-12-08 H 0.43
451 2024-12-08 T 0.43
452 2024-12-08 H 0.43
453 2024-12-09 T 0.43
454 2024-12-09 H 0.43
455 2024-12-09 H 0.43
456 2024-12-09 T 0.43
457 2024-12-09 H 0.43
458 2024-12-10 H 0.43
459 2024-12-10 T 0.43
460 2024-12-10 H 0.43
461 2024-12-10 H 0.44
462 2024-12-10 H 0.44
463 2024-12-11 T 0.44
464 2024-12-11 T 0.44
465 2024-12-11 T 0.43
466 2024-12-11 T 0.43
467 2024-12-11 H 0.43
468 2024-12-12 H 0.44
469 2024-12-12 H 0.44
470 2024-12-12 H 0.44
471 2024-12-12 T 0.44
472 2024-12-12 H 0.44
473 2024-12-13 H 0.44
474 2024-12-13 H 0.44
475 2024-12-13 H 0.44
476 2024-12-13 H 0.44
477 2024-12-13 T 0.44
478 2024-12-14 H 0.44
479 2024-12-14 H 0.44
480 2024-12-14 H 0.45
481 2024-12-14 H 0.45
482 2024-12-14 H 0.45
483 2024-12-15 H 0.45
484 2024-12-15 T 0.45
485 2024-12-15 T 0.45
486 2024-12-15 T 0.45
487 2024-12-15 H 0.45
488 2024-12-16 H 0.45
489 2024-12-16 H 0.45
490 2024-12-16 T 0.45
491 2024-12-16 T 0.45
492 2024-12-16 H 0.45
493 2024-12-17 T 0.45
494 2024-12-17 H 0.45
495 2024-12-17 T 0.45
496 2024-12-17 T 0.45
497 2024-12-17 T 0.45
498 2024-12-18 H 0.45
499 2024-12-18 H 0.45
500 2024-12-18 H 0.45
501 2024-12-18 T 0.45
502 2024-12-18 H 0.45
503 2024-12-19 T 0.45
504 2024-12-19 T 0.45
505 2024-12-19 H 0.45
506 2024-12-19 T 0.45
507 2024-12-19 T 0.45
508 2024-12-20 T 0.45
509 2024-12-20 H 0.45
510 2024-12-20 T 0.45
511 2024-12-20 H 0.45
512 2024-12-20 H 0.45
513 2024-12-21 T 0.45
514 2024-12-21 H 0.45
515 2024-12-21 T 0.45
516 2024-12-21 T 0.45
517 2024-12-21 T 0.45
518 2024-12-22 T 0.45
519 2024-12-22 T 0.45
520 2024-12-22 T 0.44
521 2024-12-22 T 0.44
522 2024-12-22 T 0.44
523 2024-12-23 T 0.44
524 2024-12-23 T 0.44
1,000People are also trading
525 2024-12-23 H 0.44
526 2024-12-23 H 0.44
527 2024-12-23 T 0.44
528 2024-12-24 H 0.44
529 2024-12-24 T 0.44
530 2024-12-24 H 0.44
531 2024-12-24 H 0.44
532 2024-12-24 T 0.44
533 2024-12-25 T 0.44
534 2024-12-25 T 0.44
535 2024-12-25 T 0.44
536 2024-12-25 H 0.44
537 2024-12-25 T 0.44
538 2024-12-26 H 0.44
539 2024-12-26 H 0.44
540 2024-12-26 H 0.44
541 2024-12-26 H 0.45
542 2024-12-26 T 0.44
543 2024-12-27 T 0.44
544 2024-12-27 H 0.44
545 2024-12-27 T 0.44
546 2024-12-27 H 0.45
547 2024-12-27 T 0.44
548 2024-12-28 T 0.44
549 2024-12-28 T 0.44
550 2024-12-28 T 0.44
551 2024-12-28 T 0.44
552 2024-12-28 T 0.44
553 2024-12-29 H 0.44
554 2024-12-29 T 0.44
555 2024-12-29 H 0.44
556 2024-12-29 T 0.44
557 2024-12-29 H 0.44
558 2024-12-30 T 0.44
559 2024-12-30 T 0.44
560 2024-12-30 T 0.44
561 2024-12-30 T 0.44
562 2024-12-30 T 0.44
563 2024-12-31 T 0.44
564 2024-12-31 T 0.44
565 2024-12-31 T 0.44
566 2024-12-31 T 0.43
567 2024-12-31 H 0.44
568 2025-01-01 H 0.44
569 2025-01-01 T 0.44
570 2025-01-01 T 0.44
571 2025-01-01 T 0.43
572 2025-01-01 T 0.43
573 2025-01-02 T 0.43
574 2025-01-02 T 0.43
575 2025-01-02 T 0.43
576 2025-01-02 H 0.43
577 2025-01-02 H 0.43
578 2025-01-03 T 0.43
579 2025-01-03 T 0.43
580 2025-01-03 H 0.43
581 2025-01-03 T 0.43
582 2025-01-03 H 0.43
583 2025-01-04 T 0.43
584 2025-01-04 T 0.43
585 2025-01-04 T 0.43
586 2025-01-04 T 0.43
587 2025-01-04 T 0.43
@gigab0nus How many flips do you expect to need to estimate p(heads) with high confidence? We could model this by randomly sampling (p(heads), seed) pairs and see how many samples are required before p_hat converges.
@biased the expectation value moved quite a bit but the fees are too high to move the price profitably.
Here's what ChatGPT o1-preview has to say:
Assessing coin bias
I'm gathering info to determine the bias in a coin by examining cumulative heads fraction after each flip. This will help in estimating the true bias towards heads or tails.
Tracking outcomes
I’m digging through the coin flips to tally heads and tails. After counting, I noted 13 heads and 7 tails. This helps in assessing the coin's bias.
Tracking outcomes
I’m mapping out the results of 20 coin flips, tallying 11 heads and 9 tails. The proportion of heads is 0.55, aligning with the table's p_hat.
Evaluating hypothesis
I’m working on determining if the coin is fair or biased by setting up null and alternative hypotheses, and using a binomial test to evaluate the probability of observing 11 or more Heads in 20 flips.
Piecing together the p-value
I’m calculating the p-value under the null hypothesis, considering the probability of observing 11 or more heads. Utilizing the symmetry of the binomial distribution, I’m focusing on cumulative probabilities to determine P(x ≥ 11) and double it due to the two-tailed nature of the test.
Calculating cumulative probabilities
I calculated cumulative probabilities using the binomial distribution's CDF, derived exact probabilities for P(x=10) and P(x=11), and determined P(x ≥ 11) by summing probabilities for x from 11 to 20.
Applying statistical methods
I’m thinking through the binomial distribution and normal approximation to find the p-value for P(X ≥ 11), ultimately concluding that the p-value is approximately 0.656, suggesting no significant bias in the coin.
Calculating the margin
I’m mapping out the margin of error for the 95% confidence interval by determining the standard error and multiplying it by the critical z-score, revealing a range from 0.332 to 0.768 for p_hat.
Based on the data provided, the coin was flipped 20 times, resulting in 11 heads and 9 tails. This gives an observed proportion (p^\hat{p}p^) of heads equal to 1120=0.55\frac{11}{20} = 0.552011=0.55.
To determine if the coin is biased, we can perform a hypothesis test:
Null Hypothesis (H0H_0H0): The coin is fair (p=0.5p = 0.5p=0.5).
Alternative Hypothesis (HaH_aHa): The coin is biased (p≠0.5p \neq 0.5p=0.5).
Calculations:
Standard Error (SE):
SE=p(1−p)n=0.5×0.520≈0.1118SE = \sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.5 \times 0.5}{20}} \approx 0.1118SE=np(1−p)=200.5×0.5≈0.1118
Test Statistic (Z-score):
Z=p^−pSE=0.55−0.50.1118≈0.447Z = \frac{\hat{p} - p}{SE} = \frac{0.55 - 0.5}{0.1118} \approx 0.447Z=SEp^−p=0.11180.55−0.5≈0.447
P-value: Using the standard normal distribution, the p-value corresponding to Z=0.447Z = 0.447Z=0.447 is approximately 0.655 (two-tailed test).
Since the p-value (0.655) is greater than the common significance level of 0.05, we fail to reject the null hypothesis.
Conclusion:
Degree of Bias: The coin shows an estimated probability of heads at 55%, but this is not statistically significant given the sample size.
Statistical Significance: There is insufficient evidence to conclude that the coin is biased.
Answer:
Based on 20 flips resulting in 11 heads and 9 tails, the coin shows no significant bias; it behaves like a fair coin with an estimated 55% chance of heads, but this deviation is within expected random variation for such a small sample.
@biased You need to tell it that the „coin“ has a uniformly randomly sampled heads-probability between 0 and 1. Then it does the analysis that I have plotted below and gives correct expectation values: 11 out of 20 => E(p)=12/22≈54.55%
The rules say that a new flip will be published for every log2(participants). However, I think this function grows too slowly relative to the general activity on Manifold. My bad, I'm new around here.
To encourage more participation in this market, I would like to propose modifying the rules to publish a new flip for every integer sqrt(volume). For example, the volume is currently 911M, so sqrt(911) = 30 flips, and the 31st flip would occur when the volume hits 961M. Here's a graph showing the volume required for each flip under this proposal:
What do people think? If there is pushback, that's fine, I can propose a new question at some point with this modified mechanic.
I also thought log2 is a bit low, I would rather ride this out until the end (>>12 flips). But publishing 24 flips now at once is a bit too much. You could publish the 7th flip at 961, making it flips=sqrt(volume)-24.
On second guess, sqrt might be a bit too fast, since the new volume required for the next flip is always 2*flips+1. And since we get new information from every flip, there might always be a reason to instantly increase volume by like 100, so we will get ~50 flips practically all at once.
Perhaps it's also worth incorporating a time component (Math.floor(daysSinceCreation) or Math.floor(weeksSinceCreation)), so that the market doesn't completely grind to a halt.
In any case I'm happy for you to modify the rules for publishing new flips as you see fit.