A game is played with an even number of 20 sided dice . You win when more then half the dice show single digit numbers , how many dice  ?

A game is played with one other player as follows : each player rolls  an even number of 20 sided dice . You win when more then half the dice show single digit numbers , you are allowed to pick how many dice you roll, how many should you roll to maximize your odds of winning ? 

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  • 3 weeks ago

    I wrote a PERL script to calculate the chances of winning:

    sub fact {

    $fact = 1;

    for (1..@_[0]) {$fact = $fact * $_;}

    return $fact;

    }

    while (1) {

    $dice = $dice + 2;

    $towin = $dice/2+1;

    $final = 0;

    for $i ($towin..$dice) {

    $chance = 0.45**$i * 0.55**($dice-$i) * fact($dice) / fact($i) / fact($dice-$i);

    $final = $final + $chance;

    }

    print "If there are $dice dice, chance of winning: $final\n";

    }

    Here's the output:

    If there are 2 dice, chance of winning: 0.2025

    If there are 4 dice, chance of winning: 0.24148125

    If there are 6 dice, chance of winning: 0.25526390625

    If there are 8 dice, chance of winning: 0.260380717382813

    If there are 10 dice, chance of winning: 0.261562700754492

    If there are 12 dice, chance of winning: 0.26068507810102

    If there are 14 dice, chance of winning: 0.25863708295181

    If there are 16 dice, chance of winning: 0.255891489454901

    If there are 18 dice, chance of winning: 0.252720328965971

    If there are 20 dice, chance of winning: 0.249289359828412

    If there are 22 dice, chance of winning: 0.245703997079663

    If there are 24 dice, chance of winning: 0.242033481965631

    If there are 26 dice, chance of winning: 0.23832441558339

    If there are 28 dice, chance of winning: 0.234608725868323

    If there are 30 dice, chance of winning: 0.230908554622066

    If there are 32 dice, chance of winning: 0.227239360138381

    If there are 34 dice, chance of winning: 0.223611945138102

    If there are 36 dice, chance of winning: 0.220033815667718

    If there are 38 dice, chance of winning: 0.216510111429805

    If there are 40 dice, chance of winning: 0.213044254669798

    If there are 42 dice, chance of winning: 0.209638410158616

    If there are 44 dice, chance of winning: 0.206293815915658

    If there are 46 dice, chance of winning: 0.203011023957625

    If there are 48 dice, chance of winning: 0.19979007745523

    If there are 50 dice, chance of winning: 0.19663064232577

    If there are 52 dice, chance of winning: 0.193532105775802

    If there are 54 dice, chance of winning: 0.190493650605279

    If there are 56 dice, chance of winning: 0.187514311557986

    If there are 58 dice, chance of winning: 0.184593018254201

    If there are 60 dice, chance of winning: 0.181728628014663

    If there are 62 dice, chance of winning: 0.178919951013853

    If there are 64 dice, chance of winning: 0.176165769575071

    If there are 66 dice, chance of winning: 0.173464852966001

    If there are 68 dice, chance of winning: 0.170815968721018

    If there are 70 dice, chance of winning: 0.168217891270839

    If there are 72 dice, chance of winning: 0.165669408477071

    If there are 74 dice, chance of winning: 0.163169326531755

    If there are 76 dice, chance of winning: 0.160716473578075

    If there are 78 dice, chance of winning: 0.158309702329293

    If there are 80 dice, chance of winning: 0.155947891902367

    If there are 82 dice, chance of winning: 0.153629949036021

    If there are 84 dice, chance of winning: 0.15135480882693

    If there are 86 dice, chance of winning: 0.149121435089503

    If there are 88 dice, chance of winning: 0.146928820422783

    If there are 90 dice, chance of winning: 0.14477598605073

    If there are 92 dice, chance of winning: 0.142661981488506

    If there are 94 dice, chance of winning: 0.140585884076648

    If there are 96 dice, chance of winning: 0.138546798416465

    If there are 98 dice, chance of winning: 0.1365438557332

    If there are 100 dice, chance of winning: 0.134576213188053

    If there are 102 dice, chance of winning: 0.13264305315585

    If there are 104 dice, chance of winning: 0.130743582481647

    If there are 106 dice, chance of winning: 0.128877031726767

    If there are 108 dice, chance of winning: 0.127042654412531

    If there are 110 dice, chance of winning: 0.125239726268125

    If there are 112 dice, chance of winning: 0.123467544487611

    If there are 114 dice, chance of winning: 0.121725426999886

    If there are 116 dice, chance of winning: 0.120012711754496

    If there are 118 dice, chance of winning: 0.118328756025392

    If there are 120 dice, chance of winning: 0.116672935734184

    If there are 122 dice, chance of winning: 0.115044644793868

    If there are 124 dice, chance of winning: 0.11344329447371

    If there are 126 dice, chance of winning: 0.111868312785562

    If there are 128 dice, chance of winning: 0.110319143891725

    If there are 130 dice, chance of winning: 0.108795247534222

    If there are 132 dice, chance of winning: 0.107296098485213

    If there are 134 dice, chance of winning: 0.105821186018184

    If there are 136 dice, chance of winning: 0.104370013399418

    If there are 138 dice, chance of winning: 0.102942097399229

    If there are 140 dice, chance of winning: 0.101536967822367

    If there are 142 dice, chance of winning: 0.100154167056975

    If there are 144 dice, chance of winning: 0.0987932496414821

    If there are 146 dice, chance of winning: 0.0974537818487626

    If there are 148 dice, chance of winning: 0.0961353412869334

    If there are 150 dice, chance of winning: 0.0948375165161373

    If there are 152 dice, chance of winning: 0.0935599066806794

    If there are 154 dice, chance of winning: 0.0923021211558901

    If there are 156 dice, chance of winning: 0.0910637792091096

    If there are 158 dice, chance of winning: 0.0898445096741984

    If there are 160 dice, chance of winning: 0.0886439506389996

    If there are 162 dice, chance of winning: 0.0874617491452002

    If there are 164 dice, chance of winning: 0.0862975609000544

    If there are 166 dice, chance of winning: 0.0851510499994531

    If there are 168 dice, chance of winning: 0.0840218886618464

    [At this point I stopped the program as it looks like the numbers are continuing to decrease.]

    From the above, it looks like you should use 10 dice for maximum chance of winning.

    • blueblood3 weeks agoReport

      I cant read that , can u write it and send an attached pic to my Yahoo mail? whitehorserandin@yahoo.com ..thanks bud

  • 3 weeks ago

    I agree with Dayton's formulation, except that his "n-1" should be "n-i".

    His probability for winning with 4 dice is almost right, but I get 0.24148, which rounds to 0.241 not 0.242.

    The probability of winning with 6 dice is around 0.255.

    The probability of winning with 8 dice is around 0.2604.

    You can keep on calculating these results, and at some point I daresay the function will turn around. It won't turn around twice IMHO, so once you find a decrease, you will have found the optimal number of dice to play with.

    Dixon's answer is simply wrong.

  • 3 weeks ago

    chance of winning with n dice played

    you will need (n/2) + 1 to have 1-9 on them.

    chance of getting single digit on individual dice = (9/20)

    sigma (i=n/2 + 1 to n) nCi (9/20)^i (11/20)^(n - 1)

    you've got to maximize that

    2 dice gives win prob of .205

    4 dice gives win prob of .242

  • Dixon
    Lv 7
    3 weeks ago

    Assuming the dice are numbered 1 to 20 the chance of any given die showing a single digit is 9/20, ie less than 0.5. Thus it is best to have as few dice as possible because this requires fewer "unlikely" outcomes.

    I don't really understand the whole other player part, if he is an opponent or you play together to win or whatever. But however it works, you want as few dice as possible, so probably 2 if that is allowed.

    • Dixon
      Lv 7
      3 weeks agoReport

      I violated the first rule of probabilities - never rely on instinct, do the calculation.

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