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擲骰子遊戲的擲骰次數表
簡介
我常被問到的一個問題是：「擲骰子遊戲裡，擲骰者撐過 x 次的機率是多少？」下表給了最多 50 次擲骰的答案。第一列是擲骰次數。第二列是該次擲骰中出現 7 點的機率。第三列是該次擲骰後仍存活的機率。
擲骰子生存表（1 至 50 回合）
| 卷 數位 | 可能性 七出局 | 可能性 生存 |
|---|---|---|
| 1 | 0.00000000 | 1.00000000 |
| 2 | 0.11111111 | 0.88888889 |
| 3 | 0.11676955 | 0.77211934 |
| 4 | 0.10476680 | 0.66735254 |
| 5 | 0.09122363 | 0.57612891 |
| 6 | 0.07891804 | 0.49721087 |
| 7 | 0.06816676 | 0.42904411 |
| 8 | 0.05885276 | 0.37019135 |
| 9 | 0.05080065 | 0.31939070 |
| 10 | 0.04384414 | 0.27554656 |
| 11 | 0.03783614 | 0.23771043 |
| 12 | 0.03264850 | 0.20506193 |
| 十三 | 0.02817002 | 0.17689190 |
| 14 | 0.02430433 | 0.15258757 |
| 15 | 0.02096801 | 0.13161956 |
| 16 | 0.01808886 | 0.11353070 |
| 17 | 0.01560445 | 0.09792625 |
| 18 | 0.01346084 | 0.08446541 |
| 19 | 0.01161138 | 0.07285403 |
| 20 | 0.01001580 | 0.06283823 |
| 21 | 0.00863931 | 0.05419892 |
| 22 | 0.00745187 | 0.04674705 |
| 23 | 0.00642755 | 0.04031950 |
| 24 | 0.00554396 | 0.03477554 |
| 二十五 | 0.00478180 | 0.02999374 |
| 二十六 | 0.00412437 | 0.02586937 |
| 二十七 | 0.00355731 | 0.02231206 |
| 二十八 | 0.00306819 | 0.01924387 |
| 二十九 | 0.00264632 | 0.01659755 |
| 三十 | 0.00228244 | 0.01431511 |
| 31 | 0.00196858 | 0.01234653 |
| 三十二 | 0.00169788 | 0.01064864 |
| 33 | 0.00146440 | 0.00918424 |
| 三十四 | 0.00126303 | 0.00792121 |
| 三十五 | 0.00108934 | 0.00683187 |
| 三十六 | 0.00093954 | 0.00589234 |
| 三十七 | 0.00081033 | 0.00508201 |
| 三十八 | 0.00069890 | 0.00438311 |
| 三十九 | 0.00060278 | 0.00378033 |
| 40 | 0.00051989 | 0.00326044 |
| 41 | 0.00044839 | 0.00281205 |
| 四十二 | 0.00038673 | 0.00242532 |
| 43 | 0.00033354 | 0.00209178 |
| 四十四 | 0.00028767 | 0.00180411 |
| 45 | 0.00024811 | 0.00155600 |
| 46 | 0.00021399 | 0.00134201 |
| 四十七 | 0.00018456 | 0.00115745 |
| 四十八 | 0.00015918 | 0.00099827 |
| 49 | 0.00013729 | 0.00086098 |
| 50 | 0.00011841 | 0.00074257 |
下表顯示了相同的訊息，但最多可容納200次投擲。由於機率非常小，因此表格採用科學計數法。
擲骰子生存表（1 至 200 次擲骰）
| 卷 數位 | 可能性 七出局 | 可能性 生存 |
|---|---|---|
| 1 | 0.000000E+00 | 1.000000E+00 |
| 2 | 1.111111E-01 | 8.888889E-01 |
| 3 | 1.167695E-01 | 7.721193E-01 |
| 4 | 1.047668E-01 | 6.673525E-01 |
| 5 | 9.122363E-02 | 5.761289E-01 |
| 6 | 7.891804E-02 | 4.972109E-01 |
| 7 | 6.816676E-02 | 4.290441E-01 |
| 8 | 5.885276E-02 | 3.701913E-01 |
| 9 | 5.080065E-02 | 3.193907E-01 |
| 10 | 4.384414E-02 | 2.755466E-01 |
| 11 | 3.783614E-02 | 2.377104E-01 |
| 12 | 3.264850E-02 | 2.050619E-01 |
| 十三 | 2.817002E-02 | 1.768919E-01 |
| 14 | 2.430433E-02 | 1.525876E-01 |
| 15 | 2.096801E-02 | 1.316196E-01 |
| 16 | 1.808886E-02 | 1.135307E-01 |
| 17 | 1.560445E-02 | 9.792625E-02 |
| 18 | 1.346084E-02 | 8.446541E-02 |
| 19 | 1.161138E-02 | 7.285403E-02 |
| 20 | 1.001580E-02 | 6.283823E-02 |
| 21 | 8.639309E-03 | 5.419892E-02 |
| 22 | 7.451869E-03 | 4.674705E-02 |
| 23 | 6.427548E-03 | 4.031950E-02 |
| 24 | 5.543963E-03 | 3.477554E-02 |
| 二十五 | 4.781795E-03 | 2.999374E-02 |
| 二十六 | 4.124373E-03 | 2.586937E-02 |
| 二十七 | 3.557310E-03 | 2.231206E-02 |
| 二十八 | 3.068195E-03 | 1.924387E-02 |
| 二十九 | 2.646317E-03 | 1.659755E-02 |
| 三十 | 2.282437E-03 | 1.431511E-02 |
| 31 | 1.968585E-03 | 1.234653E-02 |
| 三十二 | 1.697884E-03 | 1.064864E-02 |
| 33 | 1.464404E-03 | 9.184241E-03 |
| 三十四 | 1.263027E-03 | 7.921214E-03 |
| 三十五 | 1.089340E-03 | 6.831874E-03 |
| 三十六 | 9.395362E-04 | 5.892338E-03 |
| 三十七 | 8.103321E-04 | 5.082006E-03 |
| 三十八 | 6.988952E-04 | 4.383111E-03 |
| 三十九 | 6.027824E-04 | 3.780328E-03 |
| 40 | 5.198867E-04 | 3.260442E-03 |
| 41 | 4.483907E-04 | 2.812051E-03 |
| 四十二 | 3.867267E-04 | 2.425324E-03 |
| 43 | 3.335427E-04 | 2.091782E-03 |
| 四十四 | 2.876726E-04 | 1.804109E-03 |
| 45 | 2.481107E-04 | 1.555998E-03 |
| 46 | 2.139894E-04 | 1.342009E-03 |
| 四十七 | 1.845605E-04 | 1.157448E-03 |
| 四十八 | 1.591789E-04 | 9.982695E-04 |
| 49 | 1.372878E-04 | 8.609818E-04 |
| 50 | 1.184072E-04 | 7.425745E-04 |
| 51 | 1.021232E-04 | 6.404513E-04 |
| 52 | 8.807867E-05 | 5.523726E-04 |
| 53 | 7.596559E-05 | 4.764071E-04 |
| 54 | 6.551837E-05 | 4.108887E-04 |
| 55 | 5.650790E-05 | 3.543808E-04 |
| 56 | 4.873660E-05 | 3.056442E-04 |
| 57 | 4.203405E-05 | 2.636101E-04 |
| 58 | 3.625328E-05 | 2.273569E-04 |
| 59 | 3.126751E-05 | 1.960893E-04 |
| 60 | 2.696741E-05 | 1.691219E-04 |
| 61 | 2.325869E-05 | 1.458632E-04 |
| 62 | 2.006001E-05 | 1.258032E-04 |
| 63 | 1.730124E-05 | 1.085020E-04 |
| 64 | 1.492187E-05 | 9.358012E-05 |
| 65 | 1.286972E-05 | 8.071040E-05 |
| 66 | 1.109980E-05 | 6.961061E-05 |
| 67 | 9.573283E-06 | 6.003732E-05 |
| 68 | 8.256706E-06 | 5.178062E-05 |
| 69 | 7.121193E-06 | 4.465942E-05 |
| 70 | 6.141842E-06 | 3.851758E-05 |
| 71 | 5.297178E-06 | 3.322040E-05 |
| 72 | 4.568677E-06 | 2.865173E-05 |
| 73 | 3.940364E-06 | 2.471136E-05 |
| 74 | 3.398461E-06 | 2.131290E-05 |
| 75 | 2.931083E-06 | 1.838182E-05 |
| 76 | 2.527982E-06 | 1.585384E-05 |
| 77 | 2.180319E-06 | 1.367352E-05 |
| 78 | 1.880468E-06 | 1.179305E-05 |
| 79 | 1.621854E-06 | 1.017120E-05 |
| 80 | 1.398806E-06 | 8.772390E-06 |
| 81 | 1.206434E-06 | 7.565956E-06 |
| 82 | 1.040518E-06 | 6.525439E-06 |
| 83 | 8.974191E-07 | 5.628020E-06 |
| 84 | 7.740004E-07 | 4.854019E-06 |
| 85 | 6.675550E-07 | 4.186464E-06 |
| 86 | 5.757487E-07 | 3.610715E-06 |
| 87 | 4.965681E-07 | 3.114147E-06 |
| 88 | 4.282770E-07 | 2.685870E-06 |
| 89 | 3.693777E-07 | 2.316493E-06 |
| 90 | 3.185785E-07 | 1.997914E-06 |
| 91 | 2.747656E-07 | 1.723148E-06 |
| 92 | 2.369781E-07 | 1.486170E-06 |
| 93 | 2.043874E-07 | 1.281783E-06 |
| 94 | 1.762788E-07 | 1.105504E-06 |
| 95 | 1.520358E-07 | 9.534683E-07 |
| 96 | 1.311269E-07 | 8.223414E-07 |
| 97 | 1.130935E-07 | 7.092478E-07 |
| 98 | 9.754019E-08 | 6.117076E-07 |
| 99 | 8.412586E-08 | 5.275818E-07 |
| 100 | 7.255634E-08 | 4.550254E-07 |
| 101 | 6.257794E-08 | 3.924475E-07 |
| 102 | 5.397183E-08 | 3.384757E-07 |
| 103 | 4.654929E-08 | 2.919264E-07 |
| 104 | 4.014754E-08 | 2.517788E-07 |
| 105 | 3.462620E-08 | 2.171526E-07 |
| 106 | 2.986419E-08 | 1.872885E-07 |
| 107 | 2.575708E-08 | 1.615314E-07 |
| 108 | 2.221480E-08 | 1.393166E-07 |
| 109 | 1.915969E-08 | 1.201569E-07 |
| 110 | 1.652473E-08 | 1.036322E-07 |
| 111 | 1.425214E-08 | 8.938002E-08 |
| 112 | 1.229210E-08 | 7.708792E-08 |
| 113 | 1.060161E-08 | 6.648631E-08 |
| 114 | 9.143612E-09 | 5.734269E-08 |
| 115 | 7.886126E-09 | 4.945657E-08 |
| 116 | 6.801576E-09 | 4.265499E-08 |
| 117 | 5.866181E-09 | 3.678881E-08 |
| 118 | 5.059427E-09 | 3.172938E-08 |
| 119 | 4.363623E-09 | 2.736576E-08 |
| 120 | 3.763510E-09 | 2.360225E-08 |
| 121 | 3.245929E-09 | 2.035632E-08 |
| 122 | 2.799529E-09 | 1.755679E-08 |
| 123 | 2.414520E-09 | 1.514227E-08 |
| 124 | 2.082460E-09 | 1.305981E-08 |
| 125 | 1.796067E-09 | 1.126375E-08 |
| 126 | 1.549061E-09 | 9.714685E-09 |
| 127 | 1.336024E-09 | 8.378661E-09 |
| 128 | 1.152286E-09 | 7.226375E-09 |
| 129 | 9.938163E-10 | 6.232559E-09 |
| 130 | 8.571405E-10 | 5.375419E-09 |
| 131 | 7.392611E-10 | 4.636157E-09 |
| 132 | 6.375933E-10 | 3.998564E-09 |
| 133 | 5.499075E-10 | 3.448657E-09 |
| 134 | 4.742808E-10 | 2.974376E-09 |
| 135 | 4.090547E-10 | 2.565321E-09 |
| 136 | 3.527990E-10 | 2.212522E-09 |
| 137 | 3.042799E-10 | 1.908242E-09 |
| 138 | 2.624334E-10 | 1.645809E-09 |
| 139 | 2.263419E-10 | 1.419467E-09 |
| 140 | 1.952140E-10 | 1.224253E-09 |
| 141 | 1.683669E-10 | 1.055886E-09 |
| 142 | 1.452120E-10 | 9.106740E-10 |
| 143 | 1.252416E-10 | 7.854324E-10 |
| 144 | 1.080176E-10 | 6.774148E-10 |
| 145 | 9.316232E-11 | 5.842525E-10 |
| 146 | 8.035006E-11 | 5.039024E-10 |
| 147 | 6.929981E-11 | 4.346026E-10 |
| 148 | 5.976927E-11 | 3.748334E-10 |
| 149 | 5.154943E-11 | 3.232839E-10 |
| 150 | 4.446003E-11 | 2.788239E-10 |
| 151 | 3.834561E-11 | 2.404783E-10 |
| 152 | 3.307208E-11 | 2.074062E-10 |
| 153 | 2.852380E-11 | 1.788824E-10 |
| 154 | 2.460103E-11 | 1.542814E-10 |
| 155 | 2.121774E-11 | 1.330637E-10 |
| 156 | 1.829975E-11 | 1.147639E-10 |
| 157 | 1.578305E-11 | 9.898086E-11 |
| 158 | 1.361247E-11 | 8.536839E-11 |
| 159 | 1.174039E-11 | 7.362800E-11 |
| 160 | 1.012578E-11 | 6.350222E-11 |
| 161 | 8.733222E-12 | 5.476899E-11 |
| 162 | 7.532174E-12 | 4.723682E-11 |
| 163 | 6.496303E-12 | 4.074052E-11 |
| 164 | 5.602891E-12 | 3.513763E-11 |
| 165 | 4.832346E-12 | 3.030528E-11 |
| 166 | 4.167772E-12 | 2.613751E-11 |
| 167 | 3.594594E-12 | 2.254292E-11 |
| 168 | 3.100243E-12 | 1.944267E-11 |
| 169 | 2.673878E-12 | 1.676880E-11 |
| 170 | 2.306150E-12 | 1.446265E-11 |
| 171 | 1.988993E-12 | 1.247365E-11 |
| 172 | 1.715455E-12 | 1.075820E-11 |
| 173 | 1.479535E-12 | 9.278663E-12 |
| 174 | 1.276060E-12 | 8.002603E-12 |
| 175 | 1.100568E-12 | 6.902035E-12 |
| 176 | 9.492110E-13 | 5.952824E-12 |
| 177 | 8.186696E-13 | 5.134155E-12 |
| 178 | 7.060810E-13 | 4.428074E-12 |
| 179 | 6.089764E-13 | 3.819097E-12 |
| 180 | 5.252261E-13 | 3.293871E-12 |
| 181 | 4.529938E-13 | 2.840877E-12 |
| 182 | 3.906952E-13 | 2.450182E-12 |
| 183 | 3.369644E-13 | 2.113218E-12 |
| 184 | 2.906229E-13 | 1.822595E-12 |
| 185 | 2.506546E-13 | 1.571940E-12 |
| 186 | 2.161831E-13 | 1.355757E-12 |
| 187 | 1.864522E-13 | 1.169305E-12 |
| 188 | 1.608101E-13 | 1.008495E-12 |
| 189 | 1.386945E-13 | 8.698004E-13 |
| 190 | 1.196204E-13 | 7.501800E-13 |
| 191 | 1.031694E-13 | 6.470106E-13 |
| 192 | 8.898094E-14 | 5.580296E-13 |
| 193 | 7.674372E-14 | 4.812859E-13 |
| 194 | 6.618945E-14 | 4.150965E-13 |
| 195 | 5.708666E-14 | 3.580098E-13 |
| 196 | 4.923575E-14 | 3.087741E-13 |
| 197 | 4.246454E-14 | 2.663095E-13 |
| 198 | 3.662455E-14 | 2.296850E-13 |
| 199 | 3.158771E-14 | 1.980973E-13 |
| 200 | 2.724357E-14 | 1.708537E-13 |
每位投擲者的平均擲骰次數為 8.525510。
有關我如何解決這個問題，請參閱我的MathProblems.info網站，問題 204。