Calculating the probability of success in the Coupon Collector's Problem after T trials?
07-01-2011
, 10:54 PM
Quote:
There's a discussion going on about this in relation to Party's latest promotion and maybe somebody here can help us (discussion starts at this post, but the posts above may also be relevant).
As far as I can tell, you want to know the probability of collecting all 72 coupons of equal probability in T tries. That is computed by the inclusion-exclusion principle.
The probability of a PARTICULAR coupon NOT being collected in T tries is
(71/72)T.
The probability of 2 particular coupons NOT being collected in T tries is
(70/72)T.
The probability of 3 particular coupons NOT being collected in T tries is
(69/72)T.
and so on.
So the probability that AT LEAST 1 coupon is NOT collected in T tries is
72*(71/72)T - C(72,2)*(70/72)T + C(72,3)*(69/72)T - ... +
C(72,71)*(1/72)T
by inclusion-exclusion. You could use the Excel COMBIN function to compute C. You usually only have to compute the first few of these terms since the rest become small. For example, the last term above is the probability of getting the same ticket all T times. Then subtract the result from 1 to get the probability that you collect them all.
Here is an R script that computes this for T from 1 to 1000.
Code:
Tmax = 1000
coupons = 72
prob = rep(0,Tmax)
prob[1:(coupons - 1)] = 1
prob[coupons:Tmax]= 0
for (T in coupons:Tmax) {
for (k in 1:(coupons - 1)) {
prob[T] = prob[T] + (-1)^(k+1)*choose(coupons,k)*((coupons - k)/coupons)^T
}
}
prob[1:Tmax] = 1 - prob[1:Tmax]
prob
Code:
[1] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 [6] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 [11] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 [16] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 [21] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 [26] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 [31] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 [36] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 [41] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 [46] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 [51] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 [56] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 [61] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 [66] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 [71] 0.000000e+00 5.805963e-07 4.980059e-07 4.142176e-07 3.559525e-07 [76] 2.933911e-07 2.432665e-07 2.021211e-07 1.629455e-07 1.396471e-07 [81] 1.127678e-07 9.287411e-08 7.799837e-08 6.356628e-08 5.247286e-08 [86] 4.368546e-08 3.526830e-08 2.922683e-08 2.389351e-08 1.961506e-08 [91] 1.620756e-08 1.355994e-08 1.102110e-08 9.188592e-09 7.405410e-09 [96] 6.091169e-09 5.096396e-09 4.200716e-09 3.363731e-09 2.770260e-09 [101] 2.285936e-09 1.872007e-09 1.511126e-09 1.258137e-09 1.025700e-09 [106] 8.338247e-10 6.761510e-10 5.641219e-10 4.653187e-10 3.857203e-10 [111] 3.291057e-10 3.111450e-10 3.119877e-10 3.640399e-10 4.548470e-10 [116] 6.241618e-10 8.851954e-10 1.289866e-09 1.894793e-09 2.770650e-09 [121] 4.037363e-09 5.838804e-09 8.379790e-09 1.192811e-08 1.683797e-08 [126] 2.358118e-08 3.276403e-08 4.517697e-08 6.183173e-08 8.402300e-08 [131] 1.133841e-07 1.519756e-07 2.023697e-07 2.677664e-07 3.521157e-07 [136] 4.602670e-07 5.981407e-07 7.729260e-07 9.933037e-07 1.269701e-06 [141] 1.614582e-06 2.042767e-06 2.571797e-06 3.222336e-06 4.018611e-06 [146] 4.988906e-06 6.166092e-06 7.588213e-06 9.299118e-06 1.134914e-05 [151] 1.379583e-05 1.670473e-05 2.015024e-05 2.421645e-05 2.899808e-05 [156] 3.460150e-05 4.114568e-05 4.876328e-05 5.760176e-05 6.782444e-05 [161] 7.961173e-05 9.316221e-05 1.086939e-04 1.264454e-04 1.466772e-04 [166] 1.696725e-04 1.957390e-04 2.252094e-04 2.584432e-04 2.958271e-04 [171] 3.377766e-04 3.847368e-04 4.371831e-04 4.956224e-04 5.605940e-04 [176] 6.326697e-04 7.124550e-04 8.005894e-04 8.977468e-04 1.004636e-03 [181] 1.122000e-03 1.250617e-03 1.391302e-03 1.544901e-03 1.712298e-03 [186] 1.894408e-03 2.092182e-03 2.306602e-03 2.538682e-03 2.789467e-03 [191] 3.060031e-03 3.351477e-03 3.664936e-03 4.001563e-03 4.362537e-03 [196] 4.749060e-03 5.162354e-03 5.603660e-03 6.074235e-03 6.575349e-03 [201] 7.108286e-03 7.674339e-03 8.274809e-03 8.911001e-03 9.584224e-03 [206] 1.029579e-02 1.104700e-02 1.183915e-02 1.267355e-02 1.355148e-02 [211] 1.447421e-02 1.544299e-02 1.645907e-02 1.752367e-02 1.863798e-02 [216] 1.980318e-02 2.102042e-02 2.229082e-02 2.361545e-02 2.499538e-02 [221] 2.643162e-02 2.792516e-02 2.947693e-02 3.108784e-02 3.275874e-02 [226] 3.449044e-02 3.628372e-02 3.813930e-02 4.005784e-02 4.203998e-02 [231] 4.408629e-02 4.619728e-02 4.837343e-02 5.061516e-02 5.292282e-02 [236] 5.529674e-02 5.773716e-02 6.024429e-02 6.281827e-02 6.545920e-02 [241] 6.816711e-02 7.094200e-02 7.378378e-02 7.669234e-02 7.966750e-02 [246] 8.270902e-02 8.581664e-02 8.899001e-02 9.222875e-02 9.553243e-02 [251] 9.890057e-02 1.023326e-01 1.058281e-01 1.093862e-01 1.130064e-01 [256] 1.166880e-01 1.204302e-01 1.242322e-01 1.280932e-01 1.320123e-01 [261] 1.359886e-01 1.400212e-01 1.441090e-01 1.482512e-01 1.524467e-01 [266] 1.566943e-01 1.609931e-01 1.653418e-01 1.697394e-01 1.741847e-01 [271] 1.786765e-01 1.832135e-01 1.877947e-01 1.924186e-01 1.970841e-01 [276] 2.017899e-01 2.065348e-01 2.113173e-01 2.161363e-01 2.209904e-01 [281] 2.258782e-01 2.307985e-01 2.357499e-01 2.407312e-01 2.457409e-01 [286] 2.507777e-01 2.558403e-01 2.609274e-01 2.660376e-01 2.711696e-01 [291] 2.763221e-01 2.814938e-01 2.866833e-01 2.918894e-01 2.971107e-01 [296] 3.023461e-01 3.075941e-01 3.128536e-01 3.181233e-01 3.234020e-01 [301] 3.286884e-01 3.339813e-01 3.392796e-01 3.445821e-01 3.498875e-01 [306] 3.551948e-01 3.605028e-01 3.658104e-01 3.711165e-01 3.764200e-01 [311] 3.817199e-01 3.870151e-01 3.923046e-01 3.975874e-01 4.028625e-01 [316] 4.081290e-01 4.133859e-01 4.186322e-01 4.238671e-01 4.290897e-01 [321] 4.342991e-01 4.394945e-01 4.446751e-01 4.498400e-01 4.549884e-01 [326] 4.601197e-01 4.652330e-01 4.703276e-01 4.754029e-01 4.804581e-01 [331] 4.854927e-01 4.905058e-01 4.954970e-01 5.004656e-01 5.054111e-01 [336] 5.103328e-01 5.152303e-01 5.201029e-01 5.249503e-01 5.297719e-01 [341] 5.345672e-01 5.393358e-01 5.440773e-01 5.487913e-01 5.534772e-01 [346] 5.581349e-01 5.627638e-01 5.673637e-01 5.719343e-01 5.764751e-01 [351] 5.809860e-01 5.854666e-01 5.899167e-01 5.943360e-01 5.987243e-01 [356] 6.030813e-01 6.074069e-01 6.117008e-01 6.159630e-01 6.201931e-01 [361] 6.243911e-01 6.285568e-01 6.326901e-01 6.367909e-01 6.408591e-01 [366] 6.448946e-01 6.488973e-01 6.528671e-01 6.568039e-01 6.607079e-01 [371] 6.645788e-01 6.684167e-01 6.722215e-01 6.759933e-01 6.797321e-01 [376] 6.834378e-01 6.871105e-01 6.907502e-01 6.943570e-01 6.979309e-01 [381] 7.014719e-01 7.049802e-01 7.084557e-01 7.118986e-01 7.153090e-01 [386] 7.186868e-01 7.220323e-01 7.253456e-01 7.286266e-01 7.318757e-01 [391] 7.350928e-01 7.382781e-01 7.414318e-01 7.445539e-01 7.476446e-01 [396] 7.507041e-01 7.537324e-01 7.567299e-01 7.596965e-01 7.626326e-01 [401] 7.655381e-01 7.684134e-01 7.712586e-01 7.740738e-01 7.768593e-01 [406] 7.796151e-01 7.823416e-01 7.850389e-01 7.877072e-01 7.903466e-01 [411] 7.929575e-01 7.955398e-01 7.980940e-01 8.006201e-01 8.031184e-01 [416] 8.055891e-01 8.080323e-01 8.104483e-01 8.128373e-01 8.151995e-01 [421] 8.175352e-01 8.198444e-01 8.221275e-01 8.243847e-01 8.266161e-01 [426] 8.288219e-01 8.310025e-01 8.331580e-01 8.352886e-01 8.373945e-01 [431] 8.394760e-01 8.415333e-01 8.435665e-01 8.455759e-01 8.475618e-01 [436] 8.495243e-01 8.514637e-01 8.533801e-01 8.552737e-01 8.571449e-01 [441] 8.589938e-01 8.608206e-01 8.626256e-01 8.644089e-01 8.661707e-01 [446] 8.679113e-01 8.696309e-01 8.713297e-01 8.730079e-01 8.746657e-01 [451] 8.763033e-01 8.779209e-01 8.795187e-01 8.810970e-01 8.826559e-01 [456] 8.841956e-01 8.857164e-01 8.872184e-01 8.887018e-01 8.901669e-01 [461] 8.916137e-01 8.930426e-01 8.944537e-01 8.958473e-01 8.972234e-01 [466] 8.985823e-01 8.999241e-01 9.012492e-01 9.025576e-01 9.038495e-01 [471] 9.051251e-01 9.063847e-01 9.076283e-01 9.088562e-01 9.100685e-01 [476] 9.112654e-01 9.124471e-01 9.136138e-01 9.147656e-01 9.159027e-01 [481] 9.170253e-01 9.181335e-01 9.192275e-01 9.203075e-01 9.213737e-01 [486] 9.224261e-01 9.234650e-01 9.244905e-01 9.255028e-01 9.265020e-01 [491] 9.274883e-01 9.284619e-01 9.294228e-01 9.303713e-01 9.313075e-01 [496] 9.322315e-01 9.331435e-01 9.340436e-01 9.349320e-01 9.358088e-01 [501] 9.366742e-01 9.375283e-01 9.383712e-01 9.392031e-01 9.400240e-01 [506] 9.408343e-01 9.416338e-01 9.424229e-01 9.432016e-01 9.439701e-01 [511] 9.447285e-01 9.454768e-01 9.462153e-01 9.469441e-01 9.476632e-01 [516] 9.483729e-01 9.490731e-01 9.497641e-01 9.504459e-01 9.511187e-01 [521] 9.517826e-01 9.524377e-01 9.530840e-01 9.537218e-01 9.543511e-01 [526] 9.549721e-01 9.555847e-01 9.561892e-01 9.567857e-01 9.573742e-01 [531] 9.579548e-01 9.585277e-01 9.590930e-01 9.596507e-01 9.602009e-01 [536] 9.607438e-01 9.612794e-01 9.618078e-01 9.623291e-01 9.628435e-01 [541] 9.633509e-01 9.638516e-01 9.643455e-01 9.648328e-01 9.653135e-01 [546] 9.657878e-01 9.662557e-01 9.667172e-01 9.671726e-01 9.676219e-01 [551] 9.680650e-01 9.685022e-01 9.689335e-01 9.693590e-01 9.697788e-01 [556] 9.701929e-01 9.706013e-01 9.710043e-01 9.714018e-01 9.717939e-01 [561] 9.721808e-01 9.725623e-01 9.729388e-01 9.733101e-01 9.736764e-01 [566] 9.740377e-01 9.743941e-01 9.747457e-01 9.750925e-01 9.754346e-01 [571] 9.757721e-01 9.761050e-01 9.764333e-01 9.767572e-01 9.770767e-01 [576] 9.773918e-01 9.777027e-01 9.780093e-01 9.783118e-01 9.786101e-01 [581] 9.789044e-01 9.791946e-01 9.794809e-01 9.797633e-01 9.800419e-01 [586] 9.803166e-01 9.805876e-01 9.808549e-01 9.811186e-01 9.813786e-01 [591] 9.816351e-01 9.818881e-01 9.821377e-01 9.823838e-01 9.826266e-01 [596] 9.828660e-01 9.831022e-01 9.833351e-01 9.835649e-01 9.837915e-01 [601] 9.840150e-01 9.842355e-01 9.844529e-01 9.846674e-01 9.848789e-01 [606] 9.850875e-01 9.852933e-01 9.854962e-01 9.856964e-01 9.858938e-01 [611] 9.860885e-01 9.862805e-01 9.864699e-01 9.866567e-01 9.868410e-01 [616] 9.870227e-01 9.872019e-01 9.873786e-01 9.875530e-01 9.877249e-01 [621] 9.878945e-01 9.880617e-01 9.882267e-01 9.883893e-01 9.885498e-01 [626] 9.887080e-01 9.888641e-01 9.890180e-01 9.891698e-01 9.893195e-01 [631] 9.894671e-01 9.896127e-01 9.897564e-01 9.898980e-01 9.900377e-01 [636] 9.901754e-01 9.903113e-01 9.904453e-01 9.905775e-01 9.907078e-01 [641] 9.908363e-01 9.909631e-01 9.910881e-01 9.912114e-01 9.913330e-01 [646] 9.914529e-01 9.915712e-01 9.916878e-01 9.918029e-01 9.919163e-01 [651] 9.920282e-01 9.921385e-01 9.922473e-01 9.923546e-01 9.924605e-01 [656] 9.925649e-01 9.926678e-01 9.927693e-01 9.928694e-01 9.929681e-01 [661] 9.930655e-01 9.931615e-01 9.932562e-01 9.933496e-01 9.934417e-01 [666] 9.935325e-01 9.936221e-01 9.937105e-01 9.937976e-01 9.938835e-01 [671] 9.939682e-01 9.940518e-01 9.941342e-01 9.942154e-01 9.942956e-01 [676] 9.943746e-01 9.944525e-01 9.945294e-01 9.946052e-01 9.946800e-01 [681] 9.947537e-01 9.948264e-01 9.948981e-01 9.949688e-01 9.950385e-01 [686] 9.951073e-01 9.951751e-01 9.952420e-01 9.953079e-01 9.953729e-01 [691] 9.954371e-01 9.955003e-01 9.955627e-01 9.956242e-01 9.956849e-01 [696] 9.957447e-01 9.958037e-01 9.958619e-01 9.959192e-01 9.959758e-01 [701] 9.960316e-01 9.960866e-01 9.961409e-01 9.961944e-01 9.962472e-01 [706] 9.962992e-01 9.963505e-01 9.964011e-01 9.964510e-01 9.965003e-01 [711] 9.965488e-01 9.965967e-01 9.966439e-01 9.966904e-01 9.967363e-01 [716] 9.967816e-01 9.968262e-01 9.968702e-01 9.969136e-01 9.969564e-01 [721] 9.969987e-01 9.970403e-01 9.970814e-01 9.971218e-01 9.971618e-01 [726] 9.972011e-01 9.972400e-01 9.972783e-01 9.973160e-01 9.973532e-01 [731] 9.973900e-01 9.974262e-01 9.974619e-01 9.974971e-01 9.975318e-01 [736] 9.975661e-01 9.975998e-01 9.976331e-01 9.976660e-01 9.976984e-01 [741] 9.977303e-01 9.977618e-01 9.977928e-01 9.978235e-01 9.978537e-01 [746] 9.978835e-01 9.979128e-01 9.979418e-01 9.979704e-01 9.979985e-01 [751] 9.980263e-01 9.980537e-01 9.980807e-01 9.981073e-01 9.981336e-01 [756] 9.981595e-01 9.981850e-01 9.982102e-01 9.982351e-01 9.982596e-01 [761] 9.982837e-01 9.983075e-01 9.983310e-01 9.983542e-01 9.983770e-01 [766] 9.983996e-01 9.984218e-01 9.984437e-01 9.984653e-01 9.984866e-01 [771] 9.985076e-01 9.985283e-01 9.985487e-01 9.985689e-01 9.985887e-01 [776] 9.986083e-01 9.986276e-01 9.986467e-01 9.986655e-01 9.986840e-01 [781] 9.987023e-01 9.987203e-01 9.987380e-01 9.987556e-01 9.987728e-01 [786] 9.987899e-01 9.988067e-01 9.988232e-01 9.988396e-01 9.988557e-01 [791] 9.988716e-01 9.988872e-01 9.989027e-01 9.989179e-01 9.989329e-01 [796] 9.989478e-01 9.989624e-01 9.989768e-01 9.989910e-01 9.990050e-01 [801] 9.990188e-01 9.990324e-01 9.990458e-01 9.990591e-01 9.990722e-01 [806] 9.990850e-01 9.990977e-01 9.991103e-01 9.991226e-01 9.991348e-01 [811] 9.991468e-01 9.991587e-01 9.991703e-01 9.991819e-01 9.991932e-01 [816] 9.992044e-01 9.992155e-01 9.992264e-01 9.992371e-01 9.992477e-01 [821] 9.992581e-01 9.992684e-01 9.992786e-01 9.992886e-01 9.992985e-01 [826] 9.993082e-01 9.993178e-01 9.993273e-01 9.993366e-01 9.993459e-01 [831] 9.993549e-01 9.993639e-01 9.993727e-01 9.993814e-01 9.993900e-01 [836] 9.993985e-01 9.994069e-01 9.994151e-01 9.994232e-01 9.994312e-01 [841] 9.994391e-01 9.994469e-01 9.994546e-01 9.994622e-01 9.994696e-01 [846] 9.994770e-01 9.994843e-01 9.994914e-01 9.994985e-01 9.995054e-01 [851] 9.995123e-01 9.995191e-01 9.995258e-01 9.995323e-01 9.995388e-01 [856] 9.995452e-01 9.995516e-01 9.995578e-01 9.995639e-01 9.995700e-01 [861] 9.995760e-01 9.995818e-01 9.995876e-01 9.995934e-01 9.995990e-01 [866] 9.996046e-01 9.996101e-01 9.996155e-01 9.996208e-01 9.996261e-01 [871] 9.996313e-01 9.996364e-01 9.996415e-01 9.996464e-01 9.996513e-01 [876] 9.996562e-01 9.996610e-01 9.996657e-01 9.996703e-01 9.996749e-01 [881] 9.996794e-01 9.996839e-01 9.996883e-01 9.996926e-01 9.996969e-01 [886] 9.997011e-01 9.997052e-01 9.997093e-01 9.997133e-01 9.997173e-01 [891] 9.997212e-01 9.997251e-01 9.997289e-01 9.997327e-01 9.997364e-01 [896] 9.997401e-01 9.997437e-01 9.997472e-01 9.997508e-01 9.997542e-01 [901] 9.997576e-01 9.997610e-01 9.997643e-01 9.997676e-01 9.997708e-01 [906] 9.997740e-01 9.997771e-01 9.997802e-01 9.997833e-01 9.997863e-01 [911] 9.997893e-01 9.997922e-01 9.997951e-01 9.997979e-01 9.998007e-01 [916] 9.998035e-01 9.998062e-01 9.998089e-01 9.998116e-01 9.998142e-01 [921] 9.998168e-01 9.998193e-01 9.998218e-01 9.998243e-01 9.998267e-01 [926] 9.998291e-01 9.998315e-01 9.998339e-01 9.998362e-01 9.998384e-01 [931] 9.998407e-01 9.998429e-01 9.998451e-01 9.998472e-01 9.998493e-01 [936] 9.998514e-01 9.998535e-01 9.998555e-01 9.998575e-01 9.998595e-01 [941] 9.998615e-01 9.998634e-01 9.998653e-01 9.998672e-01 9.998690e-01 [946] 9.998708e-01 9.998726e-01 9.998744e-01 9.998761e-01 9.998779e-01 [951] 9.998796e-01 9.998812e-01 9.998829e-01 9.998845e-01 9.998861e-01 [956] 9.998877e-01 9.998892e-01 9.998908e-01 9.998923e-01 9.998938e-01 [961] 9.998953e-01 9.998967e-01 9.998982e-01 9.998996e-01 9.999010e-01 [966] 9.999023e-01 9.999037e-01 9.999050e-01 9.999064e-01 9.999077e-01 [971] 9.999089e-01 9.999102e-01 9.999115e-01 9.999127e-01 9.999139e-01 [976] 9.999151e-01 9.999163e-01 9.999174e-01 9.999186e-01 9.999197e-01 [981] 9.999208e-01 9.999219e-01 9.999230e-01 9.999241e-01 9.999251e-01 [986] 9.999262e-01 9.999272e-01 9.999282e-01 9.999292e-01 9.999302e-01 [991] 9.999312e-01 9.999321e-01 9.999331e-01 9.999340e-01 9.999349e-01 [996] 9.999358e-01 9.999367e-01 9.999376e-01 9.999384e-01 9.999393e-01
Last edited by BruceZ; 01-16-2013 at 05:45 AM.
Reason: Added R script, data, and latex formula
07-01-2011
, 11:55 PM
Quote:
There's a discussion going on about this in relation to Party's latest
promotion and maybe somebody here can help us (discussion starts at
this post, but the posts above may also be relevant).
"I want to know the odds that you have collected at least one of each coupon given a sample
size."
Thanks - Juk
promotion and maybe somebody here can help us (discussion starts at
this post, but the posts above may also be relevant).
"I want to know the odds that you have collected at least one of each coupon given a sample
size."
Thanks - Juk
I gather you want the distribution.
I have software that produces a table but no formula is shown.
(It always shows the easy formulas)
Here are the graphs and a table from n= 150 to 1000.
I am sure BruceZ will come up with the formula code.
Relative Frequencies
Cumulative frequencies
x = # of trials
Code:
x prob[X=x] prob[X<=x] 150 0.00000205 0.00001135 151 0.00000245 0.0000138 152 0.00000291 0.0000167 153 0.00000345 0.00002015 154 0.00000407 0.00002422 155 0.00000478 0.000029 156 0.0000056 0.0000346 157 0.00000654 0.00004115 158 0.00000762 0.00004876 159 0.00000884 0.0000576 160 0.00001022 0.00006782 161 0.00001179 0.00007961 162 0.00001355 0.00009316 163 0.00001553 0.00010869 164 0.00001775 0.00012645 165 0.00002023 0.00014668 166 0.000023 0.00016967 167 0.00002607 0.00019574 168 0.00002947 0.00022521 169 0.00003323 0.00025844 170 0.00003738 0.00029583 171 0.00004195 0.00033778 172 0.00004696 0.00038474 173 0.00005245 0.00043718 174 0.00005844 0.00049562 175 0.00006497 0.00056059 176 0.00007208 0.00063267 177 0.00007979 0.00071246 178 0.00008813 0.00080059 179 0.00009716 0.00089775 180 0.00010689 0.00100464 181 0.00011736 0.001122 182 0.00012862 0.00125062 183 0.00014068 0.0013913 184 0.0001536 0.0015449 185 0.0001674 0.0017123 186 0.00018211 0.00189441 187 0.00019777 0.00209218 188 0.00021442 0.0023066 189 0.00023208 0.00253868 190 0.00025078 0.00278947 191 0.00027056 0.00306003 192 0.00029145 0.00335148 193 0.00031346 0.00366494 194 0.00033663 0.00400156 195 0.00036097 0.00436254 196 0.00038652 0.00474906 197 0.00041329 0.00516235 198 0.00044131 0.00560366 199 0.00047057 0.00607423 200 0.00050111 0.00657535 201 0.00053294 0.00710829 202 0.00056605 0.00767434 203 0.00060047 0.00827481 204 0.00063619 0.008911 205 0.00067322 0.00958422 206 0.00071156 0.01029579 207 0.00075121 0.011047 208 0.00079216 0.01183915 209 0.0008344 0.01267355 210 0.00087793 0.01355148 211 0.00092273 0.01447421 212 0.00096878 0.01544299 213 0.00101608 0.01645907 214 0.0010646 0.01752367 215 0.00111431 0.01863798 216 0.0011652 0.01980318 217 0.00121724 0.02102042 218 0.00127039 0.02229082 219 0.00132464 0.02361545 220 0.00137993 0.02499538 221 0.00143624 0.02643162 222 0.00149354 0.02792516 223 0.00155177 0.02947693 224 0.00161091 0.03108784 225 0.0016709 0.03275874 226 0.00173171 0.03449044 227 0.00179328 0.03628372 228 0.00185558 0.0381393 229 0.00191855 0.04005784 230 0.00198214 0.04203998 231 0.0020463 0.04408629 232 0.00211099 0.04619728 233 0.00217615 0.04837343 234 0.00224173 0.05061516 235 0.00230767 0.05292282 236 0.00237392 0.05529674 237 0.00244042 0.05773716 238 0.00250713 0.06024429 239 0.00257398 0.06281827 240 0.00264093 0.0654592 241 0.00270791 0.06816711 242 0.00277488 0.070942 243 0.00284178 0.07378378 244 0.00290856 0.07669234 245 0.00297516 0.0796675 246 0.00304153 0.08270902 247 0.00310761 0.08581664 248 0.00317337 0.08899001 249 0.00323874 0.09222875 250 0.00330368 0.09553243 251 0.00336814 0.09890057 252 0.00343207 0.10233263 253 0.00349542 0.10582805 254 0.00355815 0.1093862 255 0.00362022 0.11300642 256 0.00368157 0.11668799 257 0.00374218 0.12043017 258 0.003802 0.12423217 259 0.00386098 0.12809315 260 0.0039191 0.13201225 261 0.00397631 0.13598857 262 0.00403259 0.14002116 263 0.00408789 0.14410905 264 0.00414219 0.14825124 265 0.00419545 0.15244669 266 0.00424765 0.15669434 267 0.00429876 0.16099309 268 0.00434875 0.16534184 269 0.00439759 0.16973943 270 0.00444527 0.1741847 271 0.00449177 0.17867648 272 0.00453706 0.18321354 273 0.00458113 0.18779466 274 0.00462395 0.19241861 275 0.00466552 0.19708413 276 0.00470582 0.20178995 277 0.00474483 0.20653478 278 0.00478255 0.21131732 279 0.00481896 0.21613629 280 0.00485407 0.22099035 281 0.00488785 0.2258782 282 0.00492031 0.23079851 283 0.00495144 0.23574995 284 0.00498123 0.24073118 285 0.0050097 0.24574088 286 0.00503682 0.2507777 287 0.00506262 0.25584032 288 0.00508708 0.26092739 289 0.00511021 0.2660376 290 0.00513201 0.27116962 291 0.0051525 0.27632211 292 0.00517166 0.28149378 293 0.00518952 0.2866833 294 0.00520608 0.29188939 295 0.00522135 0.29711074 296 0.00523534 0.30234607 297 0.00524805 0.30759412 298 0.0052595 0.31285362 299 0.0052697 0.31812332 300 0.00527867 0.32340199 301 0.00528641 0.32868839 302 0.00529294 0.33398134 303 0.00529828 0.33927962 304 0.00530244 0.34458206 305 0.00530543 0.34988749 306 0.00530728 0.35519477 307 0.00530799 0.36050276 308 0.0053076 0.36581036 309 0.0053061 0.37111646 310 0.00530353 0.37641999 311 0.00529989 0.38171988 312 0.00529521 0.38701509 313 0.00528951 0.39230461 314 0.00528281 0.39758742 315 0.00527512 0.40286254 316 0.00526647 0.408129 317 0.00525687 0.41338587 318 0.00524634 0.41863222 319 0.00523491 0.42386713 320 0.0052226 0.42908973 321 0.00520942 0.43429914 322 0.00519539 0.43949454 323 0.00518054 0.44467508 324 0.00516489 0.44983997 325 0.00514845 0.45498842 326 0.00513125 0.46011967 327 0.00511331 0.46523298 328 0.00509465 0.47032763 329 0.00507528 0.47540291 330 0.00505523 0.48045814 331 0.00503452 0.48549265 332 0.00501317 0.49050582 333 0.00499119 0.49549701 334 0.00496861 0.50046562 335 0.00494545 0.50541108 336 0.00492173 0.51033281 337 0.00489746 0.51523027 338 0.00487267 0.52010294 339 0.00484737 0.52495031 340 0.00482158 0.52977189 341 0.00479533 0.53456722 342 0.00476862 0.53933584 343 0.00474148 0.54407733 344 0.00471393 0.54879126 345 0.00468598 0.55347724 346 0.00465765 0.55813488 347 0.00462895 0.56276383 348 0.00459991 0.56736374 349 0.00457054 0.57193428 350 0.00454085 0.57647513 351 0.00451087 0.580986 352 0.00448061 0.58546661 353 0.00445008 0.58991669 354 0.00441929 0.59433598 355 0.00438828 0.59872426 356 0.00435704 0.60308129 357 0.00432559 0.60740688 358 0.00429395 0.61170083 359 0.00426213 0.61596296 360 0.00423014 0.6201931 361 0.004198 0.6243911 362 0.00416572 0.62855683 363 0.00413332 0.63269015 364 0.0041008 0.63679095 365 0.00406818 0.64085912 366 0.00403546 0.64489458 367 0.00400267 0.64889725 368 0.00396981 0.65286706 369 0.00393689 0.65680394 370 0.00390392 0.66070787 371 0.00387092 0.66457879 372 0.00383789 0.66841668 373 0.00380485 0.67222153 374 0.0037718 0.67599333 375 0.00373875 0.67973209 376 0.00370572 0.68343781 377 0.00367271 0.68711051 378 0.00363972 0.69075024 379 0.00360678 0.69435702 380 0.00357388 0.6979309 381 0.00354103 0.70147193 382 0.00350825 0.70498018 383 0.00347554 0.70845571 384 0.0034429 0.71189861 385 0.00341034 0.71530895 386 0.00337787 0.71868683 387 0.00334551 0.72203233 388 0.00331324 0.72534557 389 0.00328108 0.72862665 390 0.00324904 0.73187569 391 0.00321711 0.7350928 392 0.00318531 0.73827811 393 0.00315365 0.74143176 394 0.00312211 0.74455387 395 0.00309072 0.74764459 396 0.00305947 0.75070407 397 0.00302838 0.75373244 398 0.00299743 0.75672988 399 0.00296665 0.75969653 400 0.00293603 0.76263255 401 0.00290557 0.76553812 402 0.00287528 0.7684134 403 0.00284517 0.77125857 404 0.00281523 0.7740738 405 0.00278547 0.77685926 406 0.00275589 0.77961515 407 0.00272649 0.78234164 408 0.00269729 0.78503893 409 0.00266827 0.7877072 410 0.00263945 0.79034665 411 0.00261082 0.79295746 412 0.00258238 0.79553985 413 0.00255415 0.798094 414 0.00252612 0.80062011 415 0.00249829 0.8031184 416 0.00247066 0.80558906 417 0.00244324 0.8080323 418 0.00241602 0.81044832 419 0.00238901 0.81283733 420 0.00236222 0.81519955 421 0.00233563 0.81753518 422 0.00230925 0.81984443 423 0.00228309 0.82212752 424 0.00225714 0.82438466 425 0.0022314 0.82661606 426 0.00220588 0.82882194 427 0.00218057 0.83100251 428 0.00215548 0.83315798 429 0.0021306 0.83528858 430 0.00210593 0.83739451 431 0.00208149 0.839476 432 0.00205726 0.84153326 433 0.00203324 0.8435665 434 0.00200945 0.84557595 435 0.00198586 0.84756181 436 0.0019625 0.84952431 437 0.00193934 0.85146365 438 0.00191641 0.85338006 439 0.00189369 0.85527375 440 0.00187118 0.85714493 441 0.00184889 0.85899382 442 0.00182681 0.86082063 443 0.00180494 0.86262557 444 0.00178329 0.86440886 445 0.00176185 0.86617071 446 0.00174062 0.86791133 447 0.0017196 0.86963092 448 0.00169878 0.87132971 449 0.00167818 0.87300789 450 0.00165779 0.87466568 451 0.0016376 0.87630328 452 0.00161762 0.87792089 453 0.00159784 0.87951873 454 0.00157827 0.881097 455 0.0015589 0.8826559 456 0.00153973 0.88419563 457 0.00152076 0.88571639 458 0.00150199 0.88721838 459 0.00148342 0.88870181 460 0.00146505 0.89016686 461 0.00144688 0.89161374 462 0.0014289 0.89304263 463 0.00141111 0.89445374 464 0.00139352 0.89584726 465 0.00137611 0.89722337 466 0.0013589 0.89858227 467 0.00134187 0.89992415 468 0.00132504 0.90124918 469 0.00130839 0.90255757 470 0.00129192 0.90384949 471 0.00127564 0.90512513 472 0.00125954 0.90638466 473 0.00124361 0.90762828 474 0.00122787 0.90885615 475 0.00121231 0.91006846 476 0.00119692 0.91126538 477 0.00118171 0.91244709 478 0.00116667 0.91361376 479 0.0011518 0.91476557 480 0.00113711 0.91590268 481 0.00112258 0.91702526 482 0.00110822 0.91813348 483 0.00109403 0.91922752 484 0.00108 0.92030752 485 0.00106614 0.92137366 486 0.00105244 0.9224261 487 0.0010389 0.92346499 488 0.00102551 0.92449051 489 0.00101229 0.9255028 490 0.00099922 0.92650202 491 0.00098631 0.92748833 492 0.00097355 0.92846188 493 0.00096094 0.92942282 494 0.00094848 0.9303713 495 0.00093618 0.93130748 496 0.00092401 0.93223149 497 0.000912 0.93314349 498 0.00090013 0.93404362 499 0.0008884 0.93493203 500 0.00087682 0.93580885 501 0.00086538 0.93667422 502 0.00085407 0.93752829 503 0.0008429 0.93837119 504 0.00083187 0.93920307 505 0.00082098 0.94002404 506 0.00081021 0.94083425 507 0.00079958 0.94163384 508 0.00078908 0.94242292 509 0.00077871 0.94320164 510 0.00076847 0.94397011 511 0.00075836 0.94472846 512 0.00074837 0.94547683 513 0.0007385 0.94621533 514 0.00072876 0.94694408 515 0.00071913 0.94766322 516 0.00070963 0.94837285 517 0.00070025 0.9490731 518 0.00069098 0.94976408 519 0.00068183 0.95044592 520 0.0006728 0.95111872 521 0.00066388 0.9517826 522 0.00065507 0.95243767 523 0.00064637 0.95308404 524 0.00063779 0.95372183 525 0.00062931 0.95435113 526 0.00062093 0.95497207 527 0.00061267 0.95558474 528 0.00060451 0.95618925 529 0.00059645 0.9567857 530 0.0005885 0.9573742 531 0.00058065 0.95795484 532 0.00057289 0.95852774 533 0.00056524 0.95909298 534 0.00055769 0.95965067 535 0.00055023 0.96020089 536 0.00054287 0.96074376 537 0.0005356 0.96127936 538 0.00052842 0.96180778 539 0.00052134 0.96232913 540 0.00051435 0.96284348 541 0.00050745 0.96335093 542 0.00050064 0.96385157 543 0.00049392 0.96434549 544 0.00048728 0.96483277 545 0.00048073 0.9653135 546 0.00047427 0.96578777 547 0.00046789 0.96625565 548 0.00046159 0.96671724 549 0.00045537 0.96717261 550 0.00044924 0.96762185 551 0.00044318 0.96806503 552 0.00043721 0.96850224 553 0.00043131 0.96893355 554 0.00042549 0.96935904 555 0.00041974 0.96977878 556 0.00041407 0.97019285 557 0.00040848 0.97060133 558 0.00040296 0.97100429 559 0.00039751 0.9714018 560 0.00039213 0.97179393 561 0.00038682 0.97218076 562 0.00038159 0.97256234 563 0.00037642 0.97293876 564 0.00037132 0.97331008 565 0.00036629 0.97367637 566 0.00036132 0.9740377 567 0.00035642 0.97439412 568 0.00035159 0.9747457 569 0.00034682 0.97509252 570 0.00034211 0.97543463 571 0.00033746 0.97577209 572 0.00033288 0.97610497 573 0.00032835 0.97643332 574 0.00032389 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0.99993845 1000 0.00000085 0.9999393
07-02-2011
, 06:59 AM
07-02-2011
, 07:50 AM
Thanks guys!
One last thing: I thought I had found a solution to this yesterday in this thread:
http://ask.metafilter.com/164711/How...y-catch-em-all
and re-arranged to get:
p = e^(N (-e^(-M/N)))
which does seem to give approximately correct answers.
I'm guessing this is a continuous approximation - where does this come from?
Also, this quote confused me:
When you plug the value of p=0.5 into the above formula (and also when you look through your tables) you get M=~334 rather than the M=~350 I (and the person replied to in this quote in the linked thread) was expecting from the Coupon Collector's Problem formula.
From looking at sallymustang's post it also appears the M=~334 value is the median?
Why is this (ie: can you explain the above quote better)?
Juk
One last thing: I thought I had found a solution to this yesterday in this thread:
http://ask.metafilter.com/164711/How...y-catch-em-all
Quote:
Oh, of course when the number of samples is large compared to the number of slots the Poisson approximation will work. If you want to do it again at some other p of missing none, define N as the number of genes in the pool and M the number of samples you take, then
M = N ln(-N/ ln(p))
M = N ln(-N/ ln(p))
p = e^(N (-e^(-M/N)))
which does seem to give approximately correct answers.
I'm guessing this is a continuous approximation - where does this come from?
Also, this quote confused me:
Quote:
The reason why the "mean" value as given by the coupon collector's equation is incorrect is because it represents the expected number of trials after which you'd have sampled all of the pieces of DNA. This is not the same as being 99.9% certain that after you reach 35,484 trials or so, you will have sampled each piece of DNA at least once. Instead, it means that if you were to keep carrying out this problem to completion, on average, you would be finished once you reached 35,484 trials. However, for any given sampling, the real point at which you had all pieces of DNA may have been before or after 35,484 trials.
From looking at sallymustang's post it also appears the M=~334 value is the median?
Why is this (ie: can you explain the above quote better)?
Juk
07-02-2011
, 11:25 AM
Are you married with children?
Where can one find your true bio with a real photo?
hehe
It is the winstats program. The winmat is also very good. Matter of fact I think they are all very well done. Some programs lack the capability of running very large number of trials but it overall is a great tool to have in your tool box. And it is free.
(My BF best BF went to PEA) I guess that is OK if you live back east.
http://math.exeter.edu/rparris/
Rick Parris
Phillips Exeter Academy
Mathematics Department
Exeter, NH 03833
I read somewhere on his site that he uses matrices for most of the theoretical probabilities.
07-02-2011
, 11:39 AM
Quote:
Thanks guys!
One last thing: I thought I had found a solution to this yesterday in this thread:
http://ask.metafilter.com/164711/How...y-catch-em-all
and re-arranged to get:
p = e^(N (-e^(-M/N)))
which does seem to give approximately correct answers.
I'm guessing this is a continuous approximation - where does this come from?
One last thing: I thought I had found a solution to this yesterday in this thread:
http://ask.metafilter.com/164711/How...y-catch-em-all
and re-arranged to get:
p = e^(N (-e^(-M/N)))
which does seem to give approximately correct answers.
I'm guessing this is a continuous approximation - where does this come from?
We can combine the 1 with the series to get
If we replace T with M and 72 with N we get
Now when N and M become large, this is an approximate series expansion for
P(all) =~ [1 - exp(-M/N)]N
= exp{N*ln[1 - exp(-M/N)]}.
ln[1 - exp(-M/N)] =~ -exp(-M/N) for small exp(-M/N). This gives
P(all) =~ exp[N*-exp(-M/N)]
as you said. The form before this last approximation is simpler though.
Quote:
Also, this quote confused me:
When you plug the value of p=0.5 into the above formula (and also when you look through your tables) you get M=~334 rather than the M=~350 I (and the person replied to in this quote in the linked thread) was expecting from the Coupon Collector's Problem formula.
From looking at sallymustang's post it also appears the M=~334 value is the median?
Why is this (ie: can you explain the above quote better)?
When you plug the value of p=0.5 into the above formula (and also when you look through your tables) you get M=~334 rather than the M=~350 I (and the person replied to in this quote in the linked thread) was expecting from the Coupon Collector's Problem formula.
From looking at sallymustang's post it also appears the M=~334 value is the median?
Why is this (ie: can you explain the above quote better)?
72/72 + 72/71 + 72/70 + ... +72/1
=~ 350
This is because it takes 1 to get the first one (72/72), plus 72/71 to get the second one because the probability is 71/72, etc.
07-03-2011
, 09:20 PM
07-03-2011
, 11:28 PM
FYI: My Dad emailed Rick at peanut software and had a few simulations added and some other things changed.
The guy seems to be easy to contact and I am sure would know who you are and would help answer any Qs you might have about some of his theoretical probabilities formulas / matrices.
Happy 4th!