# 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 ?

### 4 Answers

- Jeff AaronLv 73 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.

- az_lenderLv 73 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.

I get the same results.

- Dayton 10538Lv 73 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

- DixonLv 73 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.

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

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