Laws of Chance Tables

Used for testing claims of success greater than what can be attributed to random chance.


How to use the tables

The column headings along the top are the number of tries that the performer gets.  The row headings along the left side are the mathematical probability of success, expressed in percent and as an equivalent fraction.  The values within the table are the range of the number of successes that can be attributed to random chance.  If, after completing all tries, the number of successes achieved is outside the range shown in the table entry, then the success can be attributed to the performer's special skill or "magical" ability.

A table entry of the form "0-b" includes zero successes within the range of random chance.  The test is to see if the number of successes falls above b.

A table entry of the form "a-b" allows testing for both unusual success and unusual failure.  If the number of successes falls below a, you could conclude that there is something amiss with the mechanisms of the test or with the performer, or that there is "anti-magic" in the air.

A table entry of "-" means that the number of tries is insufficient to test a claim of that probability.  That is, it is within the range of random chance to be successful on all tries, and therefore a test with more tries is needed to adequately test the claim.


An example

The mysterious Jerome claims that he can name the suit of any ordinary playing card without seeing it, with a success rate significantly better than random chance.  You are called upon to test Jerome's claim.  First, you construct a preliminary test without too many tries, and you consult Table I below.  The mathematical probability of guessing a suit correctly is 1 in 4, or 25%.  You could run a test with as few as 5 tries.  The entry for 5 tries at 25% probability is "0-4", so Jerome would have to get all 5 guesses correct.  Jerome might find that a lot to ask of him.  A more friendly test would be 10 tries, which as you can see from the table would require Jerome to get more than 6 guesses correct.  To pass the preliminary test of his claim, he would need to get at least 7 successes out of 10 tries.

Jerome scores 7 successes!  Maybe this guy is for real, but 1 out of 100 preliminary tests will yield a false positive.  You now construct a final test and consult Table II below.  You decide to go with 20 tries.  The entry is "0-13", so Jerome must get at least 14 successes out of 20 tries to prove his ability conclusively.  When you run the test, Jerome names the card suit correctly only 4 times out of 20, slightly less than the most likely number of successes 5.  For now, Jerome's claim of power goes unproven.


Table I.  Range of Number of Successes Attributable to Random Chance, for Preliminary Testing.

Chance of number of successes being outside the range shown:  less than 1 in 100.  If the range is shown as simply "0", then there is less than a 1 in 100 chance of getting any successes in that number of tries.

Probability in % Probability as a fraction 5 Tries 10 15 20 30 40 50 60 80 100 150 200 300 400 500 600 800 1000
0.1 1/1000 0 0 0-1 0-1 0-1 0-1 0-1 0-1 0-1 0-1 0-2 0-2 0-2 0-2 0-3 0-3 0-3 0-4
0.2 1/500 0 0-1 0-1 0-1 0-1 0-1 0-1 0-1 0-2 0-2 0-2 0-2 0-3 0-3 0-4 0-4 0-5 0-6
0.5 1/200 0-1 0-1 0-1 0-1 0-1 0-2 0-2 0-2 0-2 0-3 0-3 0-4 0-5 0-6 0-7 0-8 0-9 0-11
1.0 1/100 0-1 0-1 0-1 0-2 0-2 0-2 0-3 0-3 0-3 0-4 0-5 0-6 0-8 0-9 0-11 0-12 1-15 2-18
2.0 1/50 0-1 0-2 0-2 0-2 0-3 0-3 0-4 0-4 0-5 0-6 0-8 0-9 0-12 1-15 2-18 3-21 6-26 9-31
3.0 3/100 0-1 0-2 0-2 0-3 0-4 0-4 0-5 0-5 0-7 0-8 0-10 0-12 1-17 3-21 5-25 7-29 12-36 16-44
4.0 1/25 0-2 0-2 0-3 0-3 0-4 0-5 0-6 0-6 0-8 0-9 0-12 1-15 3-21 6-26 9-31 12-36 18-46 24-56
5.0 1/20 0-2 0-3 0-3 0-4 0-5 0-6 0-7 0-7 0-9 0-11 1-15 2-18 5-25 9-31 12-38 16-44 24-56 32-68
6.3 1/16 0-2 0-3 0-4 0-4 0-5 0-7 0-8 0-9 0-11 0-12 1-17 4-22 8-30 13-37 17-45 23-53 32-68 43-83
8.3 1/12 0-2 0-3 0-4 0-5 0-7 0-8 0-9 0-11 1-13 1-15 4-22 7-27 13-37 19-47 26-58 33-67 47-87 60-106
10.0 1/10 0-2 0-4 0-5 0-6 0-7 0-9 0-10 0-12 1-15 2-18 6-24 9-31 17-43 25-55 33-67 41-79 58-102 76-124
12.5 1/8 0-3 0-4 0-5 0-6 0-8 0-10 0-12 2-14 2-18 5-21 9-29 13-37 23-53 33-67 44-82 54-96 76-124 98-152
14.3 1/7 0-3 0-4 0-6 0-7 0-9 0-12 1-13 2-16 3-19 5-23 10-32 16-42 27-59 39-75 51-91 64-108 88-140 115-171
16.7 1/6 0-3 0-5 0-6 0-8 0-10 1-13 1-15 3-17 4-22 8-26 13-37 19-47 33-67 48-86 61-105 77-123 106-160 137-197
20.0 1/5 0-3 0-5 0-7 0-8 0-12 2-14 3-17 4-20 7-25 10-30 17-43 25-55 42-78 59-101 77-123 95-145 131-189 167-233
25.0 1/4 0-4 0-6 0-8 0-10 2-14 3-17 5-21 6-24 10-30 14-36 24-52 34-66 56-94 78-122 100-150 123-177 168-232 215-285
30.0 3/10 0-4 0-7 1-9 1-11 3-15 5-19 7-23 9-27 14-34 18-42 31-59 43-77 70-110 96-144 124-176 151-209 207-273 263-337
33.3 1/3 0-4 0-7 0-10 2-12 3-17 5-21 9-25 11-29 16-38 21-45 35-65 50-84 79-121 109-157 140-194 170-230 233-301 295-371
40.0 2/5 - 0-8 1-11 2-14 5-19 8-24 11-29 14-34 21-43 27-53 45-75 62-98 98-142 135-185 172-228 209-271 284-356 360-440
50.0 1/2 - 1-9 3-13 4-16 8-22 12-28 16-34 20-40 29-51 37-63 59-91 82-118 128-172 174-226 221-279 268-332 364-436 459-541
60.0 3/5 - - 4-14 6-18 11-25 16-32 21-39 26-46 37-59 47-73 75-105 102-138 158-202 215-265 272-328 329-391 444-516 560-640
66.7 2/3 - - - 8-18 13-27 19-35 25-41 31-49 42-64 55-79 85-115 116-150 179-221 243-291 306-360 370-430 499-567 629-705
70.0 7/10 - - - 9-19 15-27 21-35 27-43 33-51 46-66 58-82 91-119 123-157 190-230 256-304 324-376 391-449 527-593 663-737
75.0 3/4 - - - - 17-29 23-37 30-46 36-54 50-70 64-86 99-127 134-166 206-244 278-322 350-400 423-477 568-632 715-785
80.0 4/5 - - - - - 26-38 33-47 40-56 55-73 70-90 107-133 145-175 222-258 299-341 377-423 455-505 611-669 767-833
87.5 7/8 - - - - - - - 46-59 62-78 79-97 121-141 163-187 248-278 333-367 419-457 504-546 676-724 848-902
90.0 9/10 - - - - - - - - 65-79 82-98 126-144 169-191 257-283 345-375 433-467 521-559 698-742 876-924

 

Table II.  Range of Number of Successes Attributable to Random Chance, for Final Testing.

Chance of number of successes being outside the range shown:  less than 1 in 10,000.

Probability in % Probability as a fraction 5 Tries 10 15 20 30 40 50 60 80 100 150 200 300 400 500 600 800 1000
0.1 1/1000 0-1 0-1 0-2 0-2 0-2 0-2 0-2 0-2 0-2 0-3 0-3 0-3 0-4 0-4 0-5 0-5 0-6 0-6
0.2 1/500 0-1 0-2 0-2 0-2 0-2 0-2 0-3 0-3 0-3 0-3 0-4 0-4 0-5 0-6 0-6 0-7 0-8 0-9
0.5 1/200 0-2 0-2 0-2 0-3 0-3 0-3 0-4 0-4 0-4 0-5 0-6 0-6 0-8 0-9 0-10 0-11 0-13 0-15
1.0 1/100 0-2 0-3 0-3 0-3 0-4 0-4 0-5 0-5 0-6 0-6 0-8 0-9 0-11 0-13 0-15 0-17 0-20 0-24
2.0 1/50 0-2 0-3 0-4 0-4 0-5 0-6 0-6 0-7 0-8 0-9 0-11 0-13 0-17 0-20 0-23 0-27 2-33 4-38
3.0 3/100 0-3 0-4 0-4 0-5 0-6 0-7 0-8 0-8 0-10 0-11 0-14 0-17 0-22 0-27 2-31 3-35 6-44 10-52
4.0 1/25 0-3 0-4 0-5 0-5 0-7 0-8 0-9 0-10 0-11 0-13 0-17 0-20 0-26 2-32 4-38 6-44 11-55 17-65
5.0 1/20 0-3 0-4 0-5 0-6 0-7 0-9 0-10 0-11 0-13 0-15 0-19 0-23 2-31 4-38 7-45 10-52 17-65 23-78
6.3 1/16 0-3 0-5 0-6 0-7 0-8 0-10 0-11 0-12 0-15 0-17 0-22 1-27 4-36 7-45 11-53 15-61 24-77 33-93
8.3 1/12 0-4 0-5 0-6 0-8 0-10 0-11 0-13 0-14 0-17 0-20 1-27 3-33 7-45 12-56 18-66 24-77 36-98 49-118
10.0 1/10 0-4 0-6 0-7 0-8 0-10 0-12 0-14 0-16 0-19 0-23 2-30 5-37 11-51 17-64 24-77 32-89 47-113 63-137
12.5 1/8 0-4 0-6 0-8 0-9 0-12 0-14 0-16 0-18 0-22 1-26 4-35 8-44 16-60 25-76 34-92 43-107 63-137 84-166
14.3 1/7 0-4 0-6 0-8 0-10 0-13 0-15 0-18 0-20 1-24 2-29 5-39 10-48 20-67 30-85 41-102 52-120 75-153 100-186
16.7 1/6 - 0-7 0-9 0-11 0-14 0-17 0-19 0-22 1-27 3-32 8-43 13-54 25-75 38-96 50-116 64-136 92-174 121-213
20.0 1/5 - 0-7 0-10 0-12 0-15 0-18 0-22 1-25 3-30 5-36 11-50 18-62 33-87 49-111 65-135 82-158 116-204 151-249
25.0 1/4 - 0-8 0-11 0-13 0-17 0-21 1-25 3-28 6-35 9-42 17-59 26-74 46-104 66-134 87-163 109-191 152-248 197-303
30.0 3/10 - 0-9 0-11 0-14 0-19 2-23 3-28 5-32 8-40 12-48 23-67 35-85 59-121 84-156 110-190 136-224 190-290 244-356
33.3 1/3 - 0-9 0-12 0-15 1-20 2-25 4-30 6-34 11-43 15-52 28-72 41-93 68-132 96-170 126-208 155-245 215-319 275-391
40.0 2/5 - - 0-13 0-16 2-22 4-28 7-33 9-39 15-49 21-59 37-83 53-107 87-153 122-198 157-243 193-287 266-374 340-460
50.0 1/2 - - 1-14 2-18 5-25 8-32 12-38 15-45 23-57 31-69 51-99 73-127 116-184 161-239 207-293 252-348 345-455 439-561
60.0 3/5 - - - - 8-28 12-36 17-43 21-51 31-65 41-79 67-113 93-147 147-213 202-278 257-343 313-407 426-534 540-660
66.7 2/3 - - - - 10-29 15-38 20-46 26-54 37-69 48-85 78-122 107-159 168-232 230-304 292-374 355-445 481-585 609-725
70.0 7/10 - - - - - 17-38 22-47 28-55 40-72 52-88 83-127 115-165 179-241 244-316 310-390 376-464 510-610 644-756
75.0 3/4 - - - - - - 25-49 32-57 45-74 58-91 92-133 126-174 196-254 266-334 337-413 409-491 552-648 697-803
80.0 4/5 - - - - - - - 35-59 50-77 64-95 100-139 138-182 213-267 289-351 365-435 442-518 596-684 751-849
87.5 7/8 - - - - - - - - - 74-99 115-146 156-192 240-284 324-375 408-466 493-557 663-737 834-916
90.0 9/10 - - - - - - - - - - 120-148 163-195 249-289 336-383 423-476 511-568 687-753 863-937

 

Table III.  Range of Number of Successes Attributable to Random Chance, for Extremely Stringent Testing.

Chance of number of successes being outside the range shown:  less than 1 in 1,000,000.  I have heard, but have not confirmed, that this is the level of testing required for a 'formal test' in Randi's million dollar challenge.

Probability in % Probability as a fraction 5 Tries 10 15 20 30 40 50 60 80 100 150 200 300 400 500 600 800 1000
0.1 1/1000 0-2 0-2 0-2 0-3 0-3 0-3 0-3 0-3 0-4 0-4 0-4 0-5 0-5 0-6 0-6 0-7 0-8 0-9
0.2 1/500 0-2 0-2 0-3 0-3 0-3 0-4 0-4 0-4 0-4 0-5 0-5 0-6 0-7 0-8 0-9 0-9 0-10 0-12
0.5 1/200 0-3 0-3 0-3 0-4 0-4 0-5 0-5 0-5 0-6 0-6 0-8 0-8 0-10 0-12 0-13 0-14 0-17 0-19
1.0 1/100 0-3 0-4 0-4 0-5 0-5 0-6 0-6 0-7 0-8 0-8 0-10 0-12 0-14 0-16 0-19 0-21 0-24 0-28
2.0 1/50 0-3 0-4 0-5 0-6 0-7 0-8 0-8 0-9 0-10 0-11 0-14 0-16 0-21 0-24 0-28 0-31 0-38 2-44
3.0 3/100 0-4 0-5 0-6 0-7 0-8 0-9 0-10 0-11 0-12 0-14 0-17 0-20 0-26 0-31 0-36 1-41 3-50 6-59
4.0 1/25 0-4 0-5 0-6 0-7 0-9 0-10 0-11 0-12 0-14 0-16 0-20 0-24 0-31 0-38 2-44 3-50 7-62 12-73
5.0 1/20 0-4 0-6 0-7 0-8 0-10 0-11 0-12 0-14 0-16 0-18 0-23 0-27 0-36 2-44 4-51 7-58 12-73 19-86
6.3 1/16 0-4 0-6 0-8 0-9 0-11 0-12 0-14 0-15 0-18 0-20 0-26 0-32 1-41 4-51 7-60 11-69 19-86 27-102
8.3 1/12 - 0-7 0-8 0-10 0-12 0-14 0-16 0-18 0-21 0-24 0-31 0-38 4-50 9-62 14-74 19-85 30-107 42-128
10.0 1/10 - 0-7 0-9 0-10 0-13 0-15 0-17 0-19 0-23 0-27 0-35 2-43 7-57 13-71 19-85 26-98 40-123 55-148
12.5 1/8 - 0-8 0-10 0-11 0-14 0-17 0-19 0-22 0-26 0-30 2-40 4-50 12-67 20-84 28-100 37-116 56-147 74-178
14.3 1/7 - 0-8 0-10 0-12 0-15 0-18 0-21 0-23 0-28 0-33 3-44 7-54 15-74 24-93 35-111 45-129 67-164 90-198
16.7 1/6 - 0-8 0-11 0-13 0-16 0-20 0-23 0-26 0-31 1-36 5-49 9-61 20-83 32-104 44-125 56-146 82-186 110-225
20.0 1/5 - 0-9 0-12 0-14 0-18 0-22 0-25 0-28 1-35 2-41 8-55 14-69 27-95 42-120 57-145 73-169 105-216 138-262
25.0 1/4 - 0-9 0-12 0-15 0-20 0-24 0-28 1-32 3-40 6-47 13-64 21-81 39-112 58-143 78-173 98-202 140-260 183-317
30.0 3/10 - - 0-13 0-16 0-22 0-27 1-31 2-36 5-45 9-53 18-73 29-92 52-129 75-165 100-200 125-235 176-304 229-371
33.3 1/3 - - 0-14 0-17 0-23 0-28 2-33 4-38 7-48 11-57 22-79 34-100 60-140 87-179 115-219 143-257 202-332 260-406
40.0 2/5 - - - 0-18 0-25 2-31 4-37 7-42 11-53 16-64 31-89 46-114 79-161 112-208 147-253 181-299 252-388 324-476
50.0 1/2 - - - - 3-27 5-35 8-42 12-48 19-61 26-74 45-105 66-134 108-192 151-249 196-304 240-360 331-469 423-577
60.0 3/5 - - - - - 9-38 13-46 18-53 27-69 36-84 61-119 86-154 139-221 192-288 247-353 301-419 412-548 524-676
66.7 2/3 - - - - - - 17-48 22-56 32-73 43-89 71-128 100-166 160-240 221-313 281-385 343-457 468-598 594-740
70.0 7/10 - - - - - - 19-49 24-58 35-75 47-91 77-132 108-171 171-248 235-325 300-400 365-475 496-624 629-771
75.0 3/4 - - - - - - - 28-59 40-77 53-94 86-137 119-179 188-261 257-342 327-422 398-502 540-660 683-817
80.0 4/5 - - - - - - - - 45-79 59-98 95-142 131-186 205-273 280-358 355-443 431-527 584-695 738-862
87.5 7/8 - - - - - - - - - - 110-148 150-196 233-288 316-380 400-472 484-563 653-744 822-926
90.0 9/10 - - - - - - - - - - - 157-198 243-293 329-387 415-481 502-574 677-760 852-945

Last update 19-Sep-2004.  Constructed and maintained by J. Czapski, Boston, Mass.
May be freely copied and distributed.