22:
1484:
971:
127:
that is, whether the probability of the coin's falling on either side when it is tossed is exactly 50%. It is of course impossible to rule out arbitrarily small deviations from fairness such as might be expected to affect only one flip in a lifetime of flipping; also it is always possible for an unfair (or "
3530:
Determining the sex ratio in a large group of an animal species. Provided that a small random sample (i.e. small in comparison with the total population) is taken when performing the random sampling of the population, the analysis is similar to determining the probability of obtaining heads in a coin
197:
An important difference between these two approaches is that the first approach gives some weight to one's prior experience of tossing coins, while the second does not. The question of how much weight to give to prior experience, depending on the quality (credibility) of that experience, is discussed
193:). This method assumes that the experimenter can decide to toss the coin any number of times. The experimenter first decides on the level of confidence required and the tolerable margin of error. These parameters determine the minimum number of tosses that must be performed to complete the experiment.
1641:
than our presupposition of the probability that the coin was fair corresponding to the uniform prior distribution, which was 10%. Using a prior distribution that reflects our prior knowledge of what a coin is and how it acts, the posterior distribution would not favor the hypothesis of bias. However
3518:
Determining the proportion of defective items for a product subjected to a particular (but well defined) condition. Sometimes a product can be very difficult or expensive to produce. Furthermore, if testing such products will result in their destruction, a minimum number of items should be tested.
126:
is used, although the latter might be slightly "unfair" due to an asymmetrical weight distribution, which might cause one state to occur more frequently than the other, giving one party an unfair advantage. So it might be necessary to test experimentally whether the coin is in fact "fair" –
3522:
Two party polling. If a small random sample poll is taken where there are only two mutually exclusive choices, then this is similar to tossing a single coin multiple times using a possibly biased coin. A similar analysis can therefore be applied to determine the confidence to be ascribed to the
131:") coin to happen to turn up exactly 10 heads in 20 flips. Therefore, any fairness test must only establish a certain degree of confidence in a certain degree of fairness (a certain maximum bias). In more rigorous terminology, the problem is of determining the parameters of a
1850:
488:
2604:
682:
1463:
3394:
151:
Both methods prescribe an experiment (or trial) in which the coin is tossed many times and the result of each toss is recorded. The results can then be analysed statistically to decide whether the coin is "fair" or "probably not fair".
1632:
2864:
1121:
1275:
147:
This article describes experimental procedures for determining whether a coin is fair or unfair. There are many statistical methods for analyzing such an experimental procedure. This article illustrates two of them.
98:
can provide guidance on two types of question; specifically those of how many trials to undertake and of the accuracy of an estimate of the probability of turning up heads, derived from a given sample of trials.
3280:
3103:
671:
3505:
which describes the consequences of making a given decision. An approach that avoids requiring either a loss function or a prior probability (as in the
Bayesian approach) is that of "acceptance sampling".
2678:
1637:
is small when compared with the alternative hypothesis (a biased coin). However, it is not small enough to cause us to believe that the coin has a significant bias. This probability is slightly
2374:
1703:
3046:
2746:
532:) = 1. (In practice, it would be more appropriate to assume a prior distribution which is much more heavily weighted in the region around 0.5, to reflect our experience with real coins.)
294:
3483:
2513:
1929:
3446:
966:{\displaystyle f(r\mid H=h,T=t)={\frac {{N \choose h}r^{h}(1-r)^{t}}{\int _{0}^{1}{N \choose h}p^{h}(1-p)^{t}\,dp}}={\frac {r^{h}(1-r)^{t}}{\int _{0}^{1}p^{h}(1-p)^{t}\,dp}}.}
179:
which represents the information obtained from the experiment. The probability that this particular coin is a "fair coin" can then be obtained by integrating the PDF of the
3314:
2463:
3189:
3160:
3131:
2231:
1973:
2952:
2923:
2507:
94:. The practical problem of checking whether a coin is fair might be considered as easily solved by performing a sufficiently large number of trials, but statistics and
2894:
3215:
3000:
2978:
2415:
2299:
2253:
1886:
1299:
3325:
1525:
2277:
2762:
1642:
the number of trials in this example (10 tosses) is very small, and with more trials the choice of prior distribution would be somewhat less relevant.)
994:
1144:
1519:
The probability for an unbiased coin (defined for this purpose as one whose probability of coming down heads is somewhere between 45% and 55%)
272:
be the actual probability of obtaining heads in a single toss of the coin. This is the property of the coin which is being investigated. Using
163:). Initially, the true probability of obtaining a particular side when a coin is tossed is unknown, but the uncertainty is represented by the "
118:
used widely in sports and other situations where it is required to give two parties the same chance of winning. Either a specially designed
3223:
3052:
557:
90:
and, secondly, in providing a simple problem that can be used to compare various competing methods of statistical inference, including
3694:
2623:
1845:{\displaystyle \operatorname {E} =\int _{0}^{1}r\cdot f(r\mid H=7,T=3)\,\mathrm {d} r={\frac {h+1}{h+t+2}}={\frac {2}{3}}.}
3599:
However, if the coin is caught rather than allowed to bounce or spin, it is difficult to bias a coin flip's outcome. See
2319:
3679:
3665:
3647:
521:
65:
43:
183:
over the relevant interval that represents all the probabilities that can be counted as "fair" in a practical sense.
36:
3008:
2697:
483:{\displaystyle f(r\mid H=h,T=t)={\frac {\Pr(H=h\mid r,N=h+t)\,g(r)}{\int _{0}^{1}\Pr(H=h\mid p,N=h+t)\,g(p)\,dp}},}
3514:
The above mathematical analysis for determining if a coin is fair can also be applied to other uses. For example:
2599:{\displaystyle ={\sqrt {\frac {p\,(1-p)}{n}}}\leq {\sqrt {\frac {0.5\times 0.5}{n}}}={\frac {1}{2\,{\sqrt {n}}}}}
1678:
3195:
3. The coin is tossed 12000 times with a result of 5961 heads (and 6039 tails). What interval does the value of
3454:
1891:
1982:
Using this approach, to decide the number of times the coin should be tossed, two parameters are required:
1469:
211:
86:
is one whose importance lies, firstly, in providing a simple problem on which to illustrate basic ideas of
3699:
3405:
215:
3604:, Andrew; Deborah Nolan (2002). "Teacher's Corner: You Can Load a Die, But You Can't Bias a Coin".
3519:
Using a similar analysis, the probability density function of the product defect rate can be found.
30:
3291:
3217:(the true probability of obtaining heads) lie within if a confidence level of 99.999% is desired?
2424:
241:
represent more generalised variables expressing the numbers of heads and tails respectively that
3167:
3138:
3109:
2194:
1936:
3709:
3581:
3561:
2930:
2901:
2483:
180:
172:
107:
47:
2872:
1458:{\displaystyle f(r\mid H=7,T=3)={\frac {(10+1)!}{7!\,3!}}r^{7}(1-r)^{3}=1320\,r^{7}(1-r)^{3}.}
508:
The prior probability density distribution summarizes what is known about the distribution of
3389:{\displaystyle E={\frac {Z}{2\,{\sqrt {n}}}}\,={\frac {4.4172}{2\,{\sqrt {12000}}}}\,=0.0202}
548:
190:
87:
1627:{\displaystyle \Pr(0.45<r<0.55)=\int _{0.45}^{0.55}f(p\mid H=7,T=3)\,dp\approx 13\%\!}
3704:
3527:
then the analysis must take account of that, and the coin-flip analogy doesn't quite hold.)
3198:
2983:
2961:
2393:
2282:
2236:
1869:
160:
8:
3551:
2859:{\displaystyle n={\frac {Z^{2}}{4\,E^{2}}}={\frac {Z^{2}}{4\times 0.01^{2}}}=2500\ Z^{2}}
1999:
1987:
176:
3493:
Other approaches to the question of checking whether a coin is fair are available using
3621:
2262:
513:
199:
168:
164:
95:
3675:
3661:
3643:
3625:
3556:
977:
273:
132:
2756:
1. If a maximum error of 0.01 is desired, how many times should the coin be tossed?
3613:
3576:
3502:
984:
for the binomial distribution), whose denominator can be expressed in terms of the
1116:{\displaystyle f(r\mid H=h,T=t)={\frac {1}{\mathrm {B} (h+1,t+1)}}r^{h}(1-r)^{t}.}
3571:
3494:
2958:
2. If the coin is tossed 10000 times, what is the maximum error of the estimator
981:
136:
128:
91:
2309:
2003:
1998:
The confidence level is denoted by Z and is given by the Z-value of a standard
1690:
3688:
3601:
3546:
3541:
3498:
1483:
985:
111:
119:
1862:
1270:{\displaystyle f(r\mid H=h,T=t)={\frac {(h+t+1)!}{h!\,t!}}r^{h}(1-r)^{t}.}
3617:
1480:
is the probability of obtaining heads when tossing the same coin once.)
3566:
3524:
79:
3285:
Now find the value of Z corresponding to 99.999% level of confidence.
1135:
115:
103:
2304:
In statistics, the estimate of a proportion of a sample (denoted by
3275:{\displaystyle p={\frac {h}{h+t}}\,={\frac {5961}{12000}}\,=0.4968}
3098:{\displaystyle E={\frac {Z}{2\,{\sqrt {10000}}}}={\frac {Z}{200}}}
2465:. Further, in the case of a coin being tossed, it is likely that
666:{\displaystyle \Pr(H=h\mid r,N=h+t)={N \choose h}r^{h}(1-r)^{t}.}
2006:
statistics table for the normal distribution. Some examples are:
1293:= 7, i.e. the coin is tossed 10 times and 7 heads are obtained:
1645:
With the uniform prior, the posterior probability distribution
1126:
As a uniform prior distribution has been assumed, and because
3002:(the actual probability of obtaining heads in a coin toss)?
512:
in the absence of any observation. We will assume that the
501:) represents the prior probability density distribution of
123:
205:
2279:
is the same actual probability (of obtaining heads) as
114:) which are equally likely to occur. It is based on the
3497:, whose application would require the formulation of a
3523:
actual ratio of votes cast. (If people are allowed to
2673:{\displaystyle E=Z\,s_{p}={\frac {Z}{2\,{\sqrt {n}}}}}
2469:
will be not far from 0.5, so it is reasonable to take
1476:
given that 7 heads were obtained in 10 tosses. (Note:
543:
tosses of a coin with a probability of heads equal to
3457:
3408:
3328:
3294:
3226:
3201:
3170:
3141:
3112:
3055:
3011:
2986:
2964:
2933:
2904:
2875:
2765:
2700:
2626:
2613:
And hence the value of maximum error (E) is given by
2516:
2486:
2427:
2396:
2322:
2285:
2265:
2239:
2197:
1939:
1894:
1872:
1706:
1528:
1302:
1147:
997:
685:
560:
297:
3600:
2369:{\displaystyle s_{p}={\sqrt {\frac {p\,(1-p)}{n}}}}
1134:are integers, this can also be written in terms of
3477:
3440:
3388:
3308:
3274:
3209:
3183:
3154:
3125:
3097:
3040:
2994:
2972:
2946:
2917:
2888:
2858:
2740:
2672:
2598:
2501:
2457:
2409:
2368:
2293:
2271:
2247:
2225:
1967:
1923:
1880:
1844:
1626:
1457:
1269:
1115:
965:
665:
482:
3474:
3437:
3305:
3206:
2991:
2969:
2737:
2498:
2290:
2244:
1899:
1877:
1623:
622:
609:
3686:
2687:Solving for the required number of coin tosses,
1858:
1529:
561:
412:
340:
225:times and noting the observed numbers of heads,
1933:This estimator has a margin of error (E) where
2383:is the number of trials (which was denoted by
3041:{\displaystyle E={\frac {Z}{2\,{\sqrt {n}}}}}
2741:{\displaystyle n={\frac {Z^{2}}{4\,E^{2}}}\!}
815:
802:
744:
731:
676:Substituting this into the previous formula:
1677:) = 0.7; this value is called the
175:by combining the prior distribution and the
2617:
2477:
245:have been observed in the experiment. Thus
3478:{\displaystyle 0.4766<r<0.5170\,\!}
3473:
3436:
3379:
3368:
3355:
3344:
3304:
3265:
3251:
3205:
3180:
3151:
3122:
3071:
3027:
2990:
2968:
2943:
2914:
2885:
2788:
2723:
2659:
2636:
2585:
2527:
2497:
2343:
2289:
2243:
1986:The confidence level which is denoted by
1898:
1876:
1866:The best estimator for the actual value
1782:
1607:
1419:
1371:
1222:
950:
853:
467:
454:
382:
210:One method is to calculate the posterior
66:Learn how and when to remove this message
2301:of the previous section in this article.
1482:
221:A test is performed by tossing the coin
29:This article includes a list of general
3399:The interval which contains r is thus:
276:, the posterior probability density of
3687:
1924:{\displaystyle p\,\!={\frac {h}{h+t}}}
206:Posterior probability density function
157:Posterior probability density function
3660:, John Wiley & Sons, Inc. (1971)
3509:
3191:at 99.90% level of confidence (Z=3.3)
2954:at 99.90% level of confidence (Z=3.3)
1661: = 3) achieves its peak at
3642:(Example 11.7), Chapman & Hall.
2191:The maximum error (E) is defined by
1697:under the posterior distribution is
1689:. Also with the uniform prior, the
15:
3658:Introductory Engineering Statistics
3488:
3441:{\displaystyle p-E<r<p+E\,\!}
3162:at 95.45% level of confidence (Z=2)
3133:at 68.27% level of confidence (Z=1)
2925:at 95.45% level of confidence (Z=2)
2896:at 68.27% level of confidence (Z=1)
13:
3672:Data Analysis, a Bayesian Tutorial
1993:The maximum (acceptable) error (E)
1975:at a particular confidence level.
1784:
1707:
1620:
1044:
806:
735:
613:
505:, which lies in the range 0 to 1.
122:or more usually a simple currency
35:it lacks sufficient corresponding
14:
3721:
3674:, Oxford University Press (1996)
1468:The graph on the right shows the
135:, given only a limited sample of
3638:Cox, D.R., Hinkley, D.V. (1974)
1487:Plot of the probability density
20:
2002:. This value can be read off a
110:with two states (usually named
84:checking whether a coin is fair
3695:Statistical hypothesis testing
3632:
3593:
2540:
2528:
2446:
2434:
2356:
2344:
2213:
2199:
1955:
1941:
1779:
1749:
1719:
1713:
1604:
1574:
1550:
1532:
1443:
1430:
1404:
1391:
1357:
1345:
1336:
1306:
1255:
1242:
1208:
1190:
1181:
1151:
1101:
1088:
1072:
1048:
1031:
1001:
941:
928:
892:
879:
844:
831:
773:
760:
719:
689:
651:
638:
600:
564:
464:
458:
451:
415:
392:
386:
379:
343:
331:
301:
1:
3587:
1859:Estimator of true probability
535:The probability of obtaining
524:over the interval . That is,
187:Estimator of true probability
3656:Guttman, Wilks, and Hunter:
3309:{\displaystyle Z=4.4172\,\!}
2183:
2168:
2153:
2138:
2123:
2108:
2093:
2078:
2063:
2048:
2033:
2018:
2015:
2012:
1503: = 3) = 1320
1470:probability density function
212:probability density function
7:
3535:
2751:
2458:{\displaystyle p=(1-p)=0.5}
216:Bayesian probability theory
142:
10:
3726:
3184:{\displaystyle E=0.0165\,}
3155:{\displaystyle E=0.0100\,}
3126:{\displaystyle E=0.0050\,}
2387:in the previous section).
2259:of obtaining heads. Note:
2226:{\displaystyle |p-r|<E}
1968:{\displaystyle |p-r|<E}
1280:
2947:{\displaystyle n=27225\,}
2918:{\displaystyle n=10000\,}
2502:{\displaystyle s_{p}\,\!}
288:is expressed as follows:
2889:{\displaystyle n=2500\,}
2473:=0.5 in the following:
50:more precise citations.
3640:Theoretical Statistics
3582:Statistical randomness
3562:Inferential statistics
3479:
3442:
3390:
3310:
3276:
3211:
3185:
3156:
3127:
3099:
3042:
2996:
2974:
2948:
2919:
2890:
2860:
2742:
2674:
2600:
2503:
2459:
2411:
2370:
2295:
2273:
2249:
2227:
2181:% level of confidence
2166:% level of confidence
2151:% level of confidence
2136:% level of confidence
2121:% level of confidence
2106:% level of confidence
2091:% level of confidence
2076:% level of confidence
2061:% level of confidence
2046:% level of confidence
2031:% level of confidence
1969:
1925:
1882:
1846:
1628:
1516:
1459:
1271:
1117:
967:
667:
484:
181:posterior distribution
173:posterior distribution
171:is used to derive the
3606:American Statistician
3480:
3443:
3391:
3311:
3277:
3212:
3210:{\displaystyle r\,\!}
3186:
3157:
3128:
3100:
3043:
2997:
2995:{\displaystyle r\,\!}
2975:
2973:{\displaystyle p\,\!}
2949:
2920:
2891:
2861:
2743:
2675:
2601:
2504:
2460:
2412:
2410:{\displaystyle s_{p}}
2371:
2296:
2294:{\displaystyle r\,\!}
2274:
2257:estimated probability
2250:
2248:{\displaystyle p\,\!}
2228:
1970:
1926:
1883:
1881:{\displaystyle r\,\!}
1847:
1629:
1507: (1 −
1486:
1460:
1272:
1118:
968:
668:
549:binomial distribution
485:
88:statistical inference
3618:10.1198/000313002605
3455:
3406:
3326:
3292:
3224:
3199:
3168:
3139:
3110:
3053:
3009:
2984:
2962:
2931:
2902:
2873:
2763:
2698:
2624:
2514:
2484:
2425:
2394:
2390:This standard error
2320:
2283:
2263:
2237:
2195:
1937:
1892:
1870:
1704:
1526:
1300:
1145:
995:
683:
558:
295:
191:Frequentist approach
3552:Confidence interval
2000:normal distribution
1988:confidence interval
1739:
1570:
1515:ranging from 0 to 1
917:
798:
411:
177:likelihood function
112:"heads" and "tails"
3700:Bayesian inference
3510:Other applications
3475:
3438:
3386:
3306:
3272:
3207:
3181:
3152:
3123:
3095:
3038:
2992:
2970:
2944:
2915:
2886:
2856:
2738:
2670:
2596:
2499:
2455:
2407:
2366:
2291:
2269:
2245:
2223:
1965:
1921:
1878:
1842:
1725:
1624:
1556:
1517:
1455:
1267:
1113:
976:This is in fact a
963:
903:
784:
663:
514:prior distribution
480:
397:
200:credibility theory
169:Bayesian inference
165:prior distribution
108:randomizing device
96:probability theory
82:, the question of
3557:Estimation theory
3377:
3374:
3353:
3350:
3263:
3249:
3093:
3080:
3077:
3036:
3033:
2845:
2835:
2800:
2735:
2683:
2682:
2668:
2665:
2609:
2608:
2594:
2591:
2571:
2570:
2548:
2547:
2421:has a maximum at
2364:
2363:
2272:{\displaystyle r}
2188:
2187:
2016:Confidence level
1980:
1979:
1919:
1888:is the estimator
1837:
1824:
1379:
1285:For example, let
1230:
1076:
978:beta distribution
958:
861:
813:
742:
620:
475:
167:". The theory of
161:Bayesian approach
133:Bernoulli process
76:
75:
68:
3717:
3670:Devinder Sivia:
3650:
3636:
3630:
3629:
3597:
3577:Point estimation
3503:utility function
3489:Other approaches
3484:
3482:
3481:
3476:
3447:
3445:
3444:
3439:
3395:
3393:
3392:
3387:
3378:
3376:
3375:
3370:
3360:
3354:
3352:
3351:
3346:
3336:
3319:Now calculate E
3315:
3313:
3312:
3307:
3281:
3279:
3278:
3273:
3264:
3256:
3250:
3248:
3234:
3216:
3214:
3213:
3208:
3190:
3188:
3187:
3182:
3161:
3159:
3158:
3153:
3132:
3130:
3129:
3124:
3104:
3102:
3101:
3096:
3094:
3086:
3081:
3079:
3078:
3073:
3063:
3047:
3045:
3044:
3039:
3037:
3035:
3034:
3029:
3019:
3001:
2999:
2998:
2993:
2980:on the value of
2979:
2977:
2976:
2971:
2953:
2951:
2950:
2945:
2924:
2922:
2921:
2916:
2895:
2893:
2892:
2887:
2865:
2863:
2862:
2857:
2855:
2854:
2843:
2836:
2834:
2833:
2832:
2816:
2815:
2806:
2801:
2799:
2798:
2797:
2783:
2782:
2773:
2747:
2745:
2744:
2739:
2736:
2734:
2733:
2732:
2718:
2717:
2708:
2679:
2677:
2676:
2671:
2669:
2667:
2666:
2661:
2651:
2646:
2645:
2618:
2605:
2603:
2602:
2597:
2595:
2593:
2592:
2587:
2577:
2572:
2566:
2555:
2554:
2549:
2543:
2522:
2521:
2508:
2506:
2505:
2500:
2496:
2495:
2478:
2464:
2462:
2461:
2456:
2416:
2414:
2413:
2408:
2406:
2405:
2375:
2373:
2372:
2367:
2365:
2359:
2338:
2337:
2332:
2331:
2300:
2298:
2297:
2292:
2278:
2276:
2275:
2270:
2254:
2252:
2251:
2246:
2232:
2230:
2229:
2224:
2216:
2202:
2010:
2009:
1974:
1972:
1971:
1966:
1958:
1944:
1930:
1928:
1927:
1922:
1920:
1918:
1904:
1887:
1885:
1884:
1879:
1863:
1851:
1849:
1848:
1843:
1838:
1830:
1825:
1823:
1806:
1795:
1787:
1738:
1733:
1633:
1631:
1630:
1625:
1569:
1564:
1499: = 7,
1464:
1462:
1461:
1456:
1451:
1450:
1429:
1428:
1412:
1411:
1390:
1389:
1380:
1378:
1363:
1343:
1276:
1274:
1273:
1268:
1263:
1262:
1241:
1240:
1231:
1229:
1214:
1188:
1122:
1120:
1119:
1114:
1109:
1108:
1087:
1086:
1077:
1075:
1047:
1038:
972:
970:
969:
964:
959:
957:
949:
948:
927:
926:
916:
911:
901:
900:
899:
878:
877:
867:
862:
860:
852:
851:
830:
829:
820:
819:
818:
805:
797:
792:
782:
781:
780:
759:
758:
749:
748:
747:
734:
726:
672:
670:
669:
664:
659:
658:
637:
636:
627:
626:
625:
612:
547:is given by the
489:
487:
486:
481:
476:
474:
410:
405:
395:
338:
137:Bernoulli trials
106:is an idealized
71:
64:
60:
57:
51:
46:this article by
37:inline citations
24:
23:
16:
3725:
3724:
3720:
3719:
3718:
3716:
3715:
3714:
3685:
3684:
3653:
3637:
3633:
3598:
3594:
3590:
3572:Margin of error
3538:
3512:
3495:decision theory
3491:
3456:
3453:
3452:
3407:
3404:
3403:
3369:
3364:
3359:
3345:
3340:
3335:
3327:
3324:
3323:
3293:
3290:
3289:
3255:
3238:
3233:
3225:
3222:
3221:
3200:
3197:
3196:
3169:
3166:
3165:
3140:
3137:
3136:
3111:
3108:
3107:
3085:
3072:
3067:
3062:
3054:
3051:
3050:
3028:
3023:
3018:
3010:
3007:
3006:
2985:
2982:
2981:
2963:
2960:
2959:
2932:
2929:
2928:
2903:
2900:
2899:
2874:
2871:
2870:
2850:
2846:
2828:
2824:
2817:
2811:
2807:
2805:
2793:
2789:
2784:
2778:
2774:
2772:
2764:
2761:
2760:
2754:
2728:
2724:
2719:
2713:
2709:
2707:
2699:
2696:
2695:
2660:
2655:
2650:
2641:
2637:
2625:
2622:
2621:
2586:
2581:
2576:
2556:
2553:
2523:
2520:
2515:
2512:
2511:
2491:
2487:
2485:
2482:
2481:
2426:
2423:
2422:
2401:
2397:
2395:
2392:
2391:
2339:
2336:
2327:
2323:
2321:
2318:
2317:
2284:
2281:
2280:
2264:
2261:
2260:
2238:
2235:
2234:
2212:
2198:
2196:
2193:
2192:
1954:
1940:
1938:
1935:
1934:
1908:
1903:
1893:
1890:
1889:
1871:
1868:
1867:
1861:
1855:
1829:
1807:
1796:
1794:
1783:
1734:
1729:
1705:
1702:
1701:
1565:
1560:
1527:
1524:
1523:
1446:
1442:
1424:
1420:
1407:
1403:
1385:
1381:
1364:
1344:
1342:
1301:
1298:
1297:
1283:
1258:
1254:
1236:
1232:
1215:
1189:
1187:
1146:
1143:
1142:
1104:
1100:
1082:
1078:
1043:
1042:
1037:
996:
993:
992:
982:conjugate prior
944:
940:
922:
918:
912:
907:
902:
895:
891:
873:
869:
868:
866:
847:
843:
825:
821:
814:
801:
800:
799:
793:
788:
783:
776:
772:
754:
750:
743:
730:
729:
728:
727:
725:
684:
681:
680:
654:
650:
632:
628:
621:
608:
607:
606:
559:
556:
555:
406:
401:
396:
339:
337:
296:
293:
292:
280:conditional on
208:
145:
92:decision theory
72:
61:
55:
52:
42:Please help to
41:
25:
21:
12:
11:
5:
3723:
3713:
3712:
3707:
3702:
3697:
3683:
3682:
3668:
3652:
3651:
3631:
3612:(4): 308–311.
3591:
3589:
3586:
3585:
3584:
3579:
3574:
3569:
3564:
3559:
3554:
3549:
3544:
3537:
3534:
3533:
3532:
3528:
3520:
3511:
3508:
3490:
3487:
3486:
3485:
3472:
3469:
3466:
3463:
3460:
3449:
3448:
3435:
3432:
3429:
3426:
3423:
3420:
3417:
3414:
3411:
3397:
3396:
3385:
3382:
3373:
3367:
3363:
3358:
3349:
3343:
3339:
3334:
3331:
3317:
3316:
3303:
3300:
3297:
3283:
3282:
3271:
3268:
3262:
3259:
3254:
3247:
3244:
3241:
3237:
3232:
3229:
3204:
3193:
3192:
3179:
3176:
3173:
3163:
3150:
3147:
3144:
3134:
3121:
3118:
3115:
3105:
3092:
3089:
3084:
3076:
3070:
3066:
3061:
3058:
3048:
3032:
3026:
3022:
3017:
3014:
2989:
2967:
2956:
2955:
2942:
2939:
2936:
2926:
2913:
2910:
2907:
2897:
2884:
2881:
2878:
2867:
2866:
2853:
2849:
2842:
2839:
2831:
2827:
2823:
2820:
2814:
2810:
2804:
2796:
2792:
2787:
2781:
2777:
2771:
2768:
2753:
2750:
2749:
2748:
2731:
2727:
2722:
2716:
2712:
2706:
2703:
2685:
2684:
2681:
2680:
2664:
2658:
2654:
2649:
2644:
2640:
2635:
2632:
2629:
2611:
2610:
2607:
2606:
2590:
2584:
2580:
2575:
2569:
2565:
2562:
2559:
2552:
2546:
2542:
2539:
2536:
2533:
2530:
2526:
2519:
2509:
2494:
2490:
2454:
2451:
2448:
2445:
2442:
2439:
2436:
2433:
2430:
2404:
2400:
2377:
2376:
2362:
2358:
2355:
2352:
2349:
2346:
2342:
2335:
2330:
2326:
2314:
2313:
2310:standard error
2302:
2288:
2268:
2242:
2222:
2219:
2215:
2211:
2208:
2205:
2201:
2186:
2185:
2182:
2175:
2171:
2170:
2167:
2160:
2156:
2155:
2152:
2145:
2141:
2140:
2139:"Three nines"
2137:
2130:
2126:
2125:
2124:Three std dev
2122:
2115:
2111:
2110:
2107:
2100:
2096:
2095:
2092:
2085:
2081:
2080:
2077:
2070:
2066:
2065:
2062:
2055:
2051:
2050:
2047:
2040:
2036:
2035:
2032:
2025:
2021:
2020:
2017:
2014:
2008:
2007:
2004:standard score
1995:
1994:
1991:
1978:
1977:
1964:
1961:
1957:
1953:
1950:
1947:
1943:
1917:
1914:
1911:
1907:
1902:
1897:
1875:
1860:
1857:
1853:
1852:
1841:
1836:
1833:
1828:
1822:
1819:
1816:
1813:
1810:
1805:
1802:
1799:
1793:
1790:
1786:
1781:
1778:
1775:
1772:
1769:
1766:
1763:
1760:
1757:
1754:
1751:
1748:
1745:
1742:
1737:
1732:
1728:
1724:
1721:
1718:
1715:
1712:
1709:
1691:expected value
1683:(MAP) estimate
1669: / (
1635:
1634:
1622:
1619:
1616:
1613:
1610:
1606:
1603:
1600:
1597:
1594:
1591:
1588:
1585:
1582:
1579:
1576:
1573:
1568:
1563:
1559:
1555:
1552:
1549:
1546:
1543:
1540:
1537:
1534:
1531:
1466:
1465:
1454:
1449:
1445:
1441:
1438:
1435:
1432:
1427:
1423:
1418:
1415:
1410:
1406:
1402:
1399:
1396:
1393:
1388:
1384:
1377:
1374:
1370:
1367:
1362:
1359:
1356:
1353:
1350:
1347:
1341:
1338:
1335:
1332:
1329:
1326:
1323:
1320:
1317:
1314:
1311:
1308:
1305:
1282:
1279:
1278:
1277:
1266:
1261:
1257:
1253:
1250:
1247:
1244:
1239:
1235:
1228:
1225:
1221:
1218:
1213:
1210:
1207:
1204:
1201:
1198:
1195:
1192:
1186:
1183:
1180:
1177:
1174:
1171:
1168:
1165:
1162:
1159:
1156:
1153:
1150:
1124:
1123:
1112:
1107:
1103:
1099:
1096:
1093:
1090:
1085:
1081:
1074:
1071:
1068:
1065:
1062:
1059:
1056:
1053:
1050:
1046:
1041:
1036:
1033:
1030:
1027:
1024:
1021:
1018:
1015:
1012:
1009:
1006:
1003:
1000:
974:
973:
962:
956:
953:
947:
943:
939:
936:
933:
930:
925:
921:
915:
910:
906:
898:
894:
890:
887:
884:
881:
876:
872:
865:
859:
856:
850:
846:
842:
839:
836:
833:
828:
824:
817:
812:
809:
804:
796:
791:
787:
779:
775:
771:
768:
765:
762:
757:
753:
746:
741:
738:
733:
724:
721:
718:
715:
712:
709:
706:
703:
700:
697:
694:
691:
688:
674:
673:
662:
657:
653:
649:
646:
643:
640:
635:
631:
624:
619:
616:
611:
605:
602:
599:
596:
593:
590:
587:
584:
581:
578:
575:
572:
569:
566:
563:
491:
490:
479:
473:
470:
466:
463:
460:
457:
453:
450:
447:
444:
441:
438:
435:
432:
429:
426:
423:
420:
417:
414:
409:
404:
400:
394:
391:
388:
385:
381:
378:
375:
372:
369:
366:
363:
360:
357:
354:
351:
348:
345:
342:
336:
333:
330:
327:
324:
321:
318:
315:
312:
309:
306:
303:
300:
274:Bayes' theorem
233:. The symbols
207:
204:
195:
194:
184:
144:
141:
74:
73:
28:
26:
19:
9:
6:
4:
3:
2:
3722:
3711:
3710:Coin flipping
3708:
3706:
3703:
3701:
3698:
3696:
3693:
3692:
3690:
3681:
3680:0-19-851889-7
3677:
3673:
3669:
3667:
3666:0-471-33770-6
3663:
3659:
3655:
3654:
3649:
3648:0-412-12420-3
3645:
3641:
3635:
3627:
3623:
3619:
3615:
3611:
3607:
3603:
3596:
3592:
3583:
3580:
3578:
3575:
3573:
3570:
3568:
3565:
3563:
3560:
3558:
3555:
3553:
3550:
3548:
3547:Coin flipping
3545:
3543:
3542:Binomial test
3540:
3539:
3529:
3526:
3521:
3517:
3516:
3515:
3507:
3504:
3500:
3499:loss function
3496:
3470:
3467:
3464:
3461:
3458:
3451:
3450:
3433:
3430:
3427:
3424:
3421:
3418:
3415:
3412:
3409:
3402:
3401:
3400:
3383:
3380:
3371:
3365:
3361:
3356:
3347:
3341:
3337:
3332:
3329:
3322:
3321:
3320:
3301:
3298:
3295:
3288:
3287:
3286:
3269:
3266:
3260:
3257:
3252:
3245:
3242:
3239:
3235:
3230:
3227:
3220:
3219:
3218:
3202:
3177:
3174:
3171:
3164:
3148:
3145:
3142:
3135:
3119:
3116:
3113:
3106:
3090:
3087:
3082:
3074:
3068:
3064:
3059:
3056:
3049:
3030:
3024:
3020:
3015:
3012:
3005:
3004:
3003:
2987:
2965:
2940:
2937:
2934:
2927:
2911:
2908:
2905:
2898:
2882:
2879:
2876:
2869:
2868:
2851:
2847:
2840:
2837:
2829:
2825:
2821:
2818:
2812:
2808:
2802:
2794:
2790:
2785:
2779:
2775:
2769:
2766:
2759:
2758:
2757:
2729:
2725:
2720:
2714:
2710:
2704:
2701:
2694:
2693:
2692:
2690:
2662:
2656:
2652:
2647:
2642:
2638:
2633:
2630:
2627:
2620:
2619:
2616:
2615:
2614:
2588:
2582:
2578:
2573:
2567:
2563:
2560:
2557:
2550:
2544:
2537:
2534:
2531:
2524:
2517:
2510:
2492:
2488:
2480:
2479:
2476:
2475:
2474:
2472:
2468:
2452:
2449:
2443:
2440:
2437:
2431:
2428:
2420:
2402:
2398:
2388:
2386:
2382:
2360:
2353:
2350:
2347:
2340:
2333:
2328:
2324:
2316:
2315:
2311:
2307:
2303:
2286:
2266:
2258:
2240:
2220:
2217:
2209:
2206:
2203:
2190:
2189:
2184:"Five nines"
2180:
2176:
2173:
2172:
2169:Four std dev
2165:
2161:
2158:
2157:
2154:"Four nines"
2150:
2146:
2143:
2142:
2135:
2131:
2128:
2127:
2120:
2116:
2113:
2112:
2105:
2101:
2098:
2097:
2090:
2086:
2083:
2082:
2075:
2071:
2068:
2067:
2060:
2056:
2053:
2052:
2045:
2041:
2038:
2037:
2030:
2026:
2023:
2022:
2011:
2005:
2001:
1997:
1996:
1992:
1989:
1985:
1984:
1983:
1976:
1962:
1959:
1951:
1948:
1945:
1915:
1912:
1909:
1905:
1900:
1895:
1873:
1865:
1864:
1856:
1839:
1834:
1831:
1826:
1820:
1817:
1814:
1811:
1808:
1803:
1800:
1797:
1791:
1788:
1776:
1773:
1770:
1767:
1764:
1761:
1758:
1755:
1752:
1746:
1743:
1740:
1735:
1730:
1726:
1722:
1716:
1710:
1700:
1699:
1698:
1696:
1692:
1688:
1684:
1682:
1676:
1673: +
1672:
1668:
1665: =
1664:
1660:
1656:
1653: |
1652:
1648:
1643:
1640:
1617:
1614:
1611:
1608:
1601:
1598:
1595:
1592:
1589:
1586:
1583:
1580:
1577:
1571:
1566:
1561:
1557:
1553:
1547:
1544:
1541:
1538:
1535:
1522:
1521:
1520:
1514:
1510:
1506:
1502:
1498:
1495: |
1494:
1490:
1485:
1481:
1479:
1475:
1471:
1452:
1447:
1439:
1436:
1433:
1425:
1421:
1416:
1413:
1408:
1400:
1397:
1394:
1386:
1382:
1375:
1372:
1368:
1365:
1360:
1354:
1351:
1348:
1339:
1333:
1330:
1327:
1324:
1321:
1318:
1315:
1312:
1309:
1303:
1296:
1295:
1294:
1292:
1288:
1264:
1259:
1251:
1248:
1245:
1237:
1233:
1226:
1223:
1219:
1216:
1211:
1205:
1202:
1199:
1196:
1193:
1184:
1178:
1175:
1172:
1169:
1166:
1163:
1160:
1157:
1154:
1148:
1141:
1140:
1139:
1137:
1133:
1129:
1110:
1105:
1097:
1094:
1091:
1083:
1079:
1069:
1066:
1063:
1060:
1057:
1054:
1051:
1039:
1034:
1028:
1025:
1022:
1019:
1016:
1013:
1010:
1007:
1004:
998:
991:
990:
989:
987:
986:beta function
983:
979:
960:
954:
951:
945:
937:
934:
931:
923:
919:
913:
908:
904:
896:
888:
885:
882:
874:
870:
863:
857:
854:
848:
840:
837:
834:
826:
822:
810:
807:
794:
789:
785:
777:
769:
766:
763:
755:
751:
739:
736:
722:
716:
713:
710:
707:
704:
701:
698:
695:
692:
686:
679:
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660:
655:
647:
644:
641:
633:
629:
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614:
603:
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585:
582:
579:
576:
573:
570:
567:
554:
553:
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550:
546:
542:
538:
533:
531:
527:
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519:
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511:
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477:
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455:
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248:
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236:
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229:, and tails,
228:
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203:
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70:
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59:
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32:
27:
18:
17:
3671:
3657:
3639:
3634:
3609:
3605:
3595:
3513:
3492:
3398:
3318:
3284:
3194:
2957:
2755:
2688:
2686:
2612:
2470:
2466:
2418:
2417:function of
2389:
2384:
2380:
2378:
2305:
2256:
2178:
2163:
2148:
2133:
2118:
2109:"Two nines"
2103:
2094:Two std dev
2088:
2073:
2058:
2049:One std dev
2043:
2028:
1981:
1932:
1854:
1694:
1686:
1681:a posteriori
1680:
1674:
1670:
1666:
1662:
1658:
1654:
1650:
1646:
1644:
1638:
1636:
1518:
1512:
1508:
1504:
1500:
1496:
1492:
1488:
1477:
1473:
1467:
1290:
1286:
1284:
1131:
1127:
1125:
975:
675:
544:
540:
536:
534:
529:
525:
517:
509:
507:
502:
498:
494:
492:
285:
281:
277:
269:
267:
262:
258:
254:
250:
246:
242:
238:
234:
230:
226:
222:
220:
209:
196:
186:
156:
150:
146:
101:
83:
77:
62:
56:January 2010
53:
34:
3705:Experiments
3567:Loaded dice
2079:95 percent
2064:"One nine"
48:introducing
3689:Categories
3588:References
1657: = 7,
1136:factorials
268:Next, let
159:, or PDF (
80:statistics
31:references
3626:123597087
3413:−
2822:×
2561:×
2551:≤
2535:−
2441:−
2351:−
2312:given by:
2207:−
1949:−
1756:∣
1744:⋅
1727:∫
1711:
1621:%
1615:≈
1581:∣
1558:∫
1437:−
1398:−
1313:∣
1249:−
1158:∣
1095:−
1008:∣
935:−
905:∫
886:−
838:−
786:∫
767:−
696:∣
645:−
577:∣
539:heads in
428:∣
399:∫
356:∣
308:∣
116:coin flip
104:fair coin
3536:See also
2752:Examples
2308:) has a
2019:Comment
2013:Z value
1679:maximum
143:Preamble
3525:abstain
2255:is the
2174:4.4172
2159:4.0000
2144:3.8906
2129:3.2905
2114:3.0000
2099:2.5759
2084:2.0000
2069:1.9599
2054:1.6449
2039:1.0000
2024:0.6745
1511:) with
1281:Example
522:uniform
44:improve
3678:
3664:
3646:
3624:
3602:Gelman
3471:0.5170
3459:0.4766
3384:0.0202
3362:4.4172
3302:4.4172
3270:0.4968
3178:0.0165
3149:0.0100
3120:0.0050
2844:
2379:where
2233:where
2179:99.999
2177:gives
2164:99.993
2162:gives
2149:99.990
2147:gives
2134:99.900
2132:gives
2119:99.730
2117:gives
2104:99.000
2102:gives
2089:95.450
2087:gives
2074:95.000
2072:gives
2059:90.000
2057:gives
2044:68.269
2042:gives
2029:50.000
2027:gives
1639:higher
1289:= 10,
493:where
198:under
129:biased
33:, but
3622:S2CID
3531:toss.
3372:12000
3261:12000
3075:10000
2941:27225
2912:10000
2034:Half
980:(the
243:might
3676:ISBN
3662:ISBN
3644:ISBN
3468:<
3462:<
3425:<
3419:<
3258:5961
2883:2500
2841:2500
2826:0.01
2218:<
1960:<
1567:0.55
1562:0.45
1548:0.55
1545:<
1539:<
1536:0.45
1417:1320
1130:and
284:and
237:and
124:coin
120:chip
3614:doi
3501:or
3091:200
2564:0.5
2558:0.5
2453:0.5
1990:(Z)
1693:of
1685:of
1472:of
520:is
516:of
214:of
78:In
3691::
3620:.
3610:56
3608:.
2691:,
1931:.
1618:13
1530:Pr
1349:10
1138::
988::
562:Pr
551::
413:Pr
341:Pr
265:.
261:+
257:=
253:+
249:=
218:.
202:.
139:.
102:A
3628:.
3616::
3465:r
3434:E
3431:+
3428:p
3422:r
3416:E
3410:p
3381:=
3366:2
3357:=
3348:n
3342:2
3338:Z
3333:=
3330:E
3299:=
3296:Z
3267:=
3253:=
3246:t
3243:+
3240:h
3236:h
3231:=
3228:p
3203:r
3175:=
3172:E
3146:=
3143:E
3117:=
3114:E
3088:Z
3083:=
3069:2
3065:Z
3060:=
3057:E
3031:n
3025:2
3021:Z
3016:=
3013:E
2988:r
2966:p
2938:=
2935:n
2909:=
2906:n
2880:=
2877:n
2852:2
2848:Z
2838:=
2830:2
2819:4
2813:2
2809:Z
2803:=
2795:2
2791:E
2786:4
2780:2
2776:Z
2770:=
2767:n
2730:2
2726:E
2721:4
2715:2
2711:Z
2705:=
2702:n
2689:n
2663:n
2657:2
2653:Z
2648:=
2643:p
2639:s
2634:Z
2631:=
2628:E
2589:n
2583:2
2579:1
2574:=
2568:n
2545:n
2541:)
2538:p
2532:1
2529:(
2525:p
2518:=
2493:p
2489:s
2471:p
2467:p
2450:=
2447:)
2444:p
2438:1
2435:(
2432:=
2429:p
2419:p
2403:p
2399:s
2385:N
2381:n
2361:n
2357:)
2354:p
2348:1
2345:(
2341:p
2334:=
2329:p
2325:s
2306:p
2287:r
2267:r
2241:p
2221:E
2214:|
2210:r
2204:p
2200:|
1963:E
1956:|
1952:r
1946:p
1942:|
1916:t
1913:+
1910:h
1906:h
1901:=
1896:p
1874:r
1840:.
1835:3
1832:2
1827:=
1821:2
1818:+
1815:t
1812:+
1809:h
1804:1
1801:+
1798:h
1792:=
1789:r
1785:d
1780:)
1777:3
1774:=
1771:T
1768:,
1765:7
1762:=
1759:H
1753:r
1750:(
1747:f
1741:r
1736:1
1731:0
1723:=
1720:]
1717:r
1714:[
1708:E
1695:r
1687:r
1675:t
1671:h
1667:h
1663:r
1659:T
1655:H
1651:r
1649:(
1647:f
1612:p
1609:d
1605:)
1602:3
1599:=
1596:T
1593:,
1590:7
1587:=
1584:H
1578:p
1575:(
1572:f
1554:=
1551:)
1542:r
1533:(
1513:r
1509:r
1505:r
1501:T
1497:H
1493:r
1491:(
1489:f
1478:r
1474:r
1453:.
1448:3
1444:)
1440:r
1434:1
1431:(
1426:7
1422:r
1414:=
1409:3
1405:)
1401:r
1395:1
1392:(
1387:7
1383:r
1376:!
1373:3
1369:!
1366:7
1361:!
1358:)
1355:1
1352:+
1346:(
1340:=
1337:)
1334:3
1331:=
1328:T
1325:,
1322:7
1319:=
1316:H
1310:r
1307:(
1304:f
1291:h
1287:N
1265:.
1260:t
1256:)
1252:r
1246:1
1243:(
1238:h
1234:r
1227:!
1224:t
1220:!
1217:h
1212:!
1209:)
1206:1
1203:+
1200:t
1197:+
1194:h
1191:(
1185:=
1182:)
1179:t
1176:=
1173:T
1170:,
1167:h
1164:=
1161:H
1155:r
1152:(
1149:f
1132:t
1128:h
1111:.
1106:t
1102:)
1098:r
1092:1
1089:(
1084:h
1080:r
1073:)
1070:1
1067:+
1064:t
1061:,
1058:1
1055:+
1052:h
1049:(
1045:B
1040:1
1035:=
1032:)
1029:t
1026:=
1023:T
1020:,
1017:h
1014:=
1011:H
1005:r
1002:(
999:f
961:.
955:p
952:d
946:t
942:)
938:p
932:1
929:(
924:h
920:p
914:1
909:0
897:t
893:)
889:r
883:1
880:(
875:h
871:r
864:=
858:p
855:d
849:t
845:)
841:p
835:1
832:(
827:h
823:p
816:)
811:h
808:N
803:(
795:1
790:0
778:t
774:)
770:r
764:1
761:(
756:h
752:r
745:)
740:h
737:N
732:(
723:=
720:)
717:t
714:=
711:T
708:,
705:h
702:=
699:H
693:r
690:(
687:f
661:.
656:t
652:)
648:r
642:1
639:(
634:h
630:r
623:)
618:h
615:N
610:(
604:=
601:)
598:t
595:+
592:h
589:=
586:N
583:,
580:r
574:h
571:=
568:H
565:(
545:r
541:N
537:h
530:r
528:(
526:g
518:r
510:r
503:r
499:r
497:(
495:g
478:,
472:p
469:d
465:)
462:p
459:(
456:g
452:)
449:t
446:+
443:h
440:=
437:N
434:,
431:p
425:h
422:=
419:H
416:(
408:1
403:0
393:)
390:r
387:(
384:g
380:)
377:t
374:+
371:h
368:=
365:N
362:,
359:r
353:h
350:=
347:H
344:(
335:=
332:)
329:t
326:=
323:T
320:,
317:h
314:=
311:H
305:r
302:(
299:f
286:t
282:h
278:r
270:r
263:t
259:h
255:T
251:H
247:N
239:T
235:H
231:t
227:h
223:N
189:(
69:)
63:(
58:)
54:(
40:.
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