Saving Roll Result Statistics: Monte Carlo simulation

[stefanj]

A long time ago, someone on the Trollholla board posted a table that showed the chance of getting certain numbers when rolling DARO.

I lost the table and forgot the person's name, but wanted that information.

So I wrote a little Perl script which:

Rolled a DARO roll (2d6, doubles add and roll over) one million times.

Figured out how many times each result from 3 (the minimum) to 60 (enough for a LUK 10 person to make a Level 9 saving roll) would occur.

Figured out the percentage of getting each result and above.

This is a result of the run of the simulation:

Saving throw results of 3: Total generated: 55016, Percentage results: 5.5016%, Percentage of this or above: 99.9995%
Saving throw results of 4: Total generated: 55437, Percentage results: 5.5437%, Percentage of this or above: 94.4979%
Saving throw results of 5: Total generated: 113052, Percentage results: 11.3052%, Percentage of this or above: 88.9542%
Saving throw results of 6: Total generated: 112637, Percentage results: 11.2637%, Percentage of this or above: 77.649%
Saving throw results of 7: Total generated: 171779, Percentage results: 17.1779%, Percentage of this or above: 66.3853%
Saving throw results of 8: Total generated: 115397, Percentage results: 11.5397%, Percentage of this or above: 49.2074%
Saving throw results of 9: Total generated: 120647, Percentage results: 12.0647%, Percentage of this or above: 37.6677%
Saving throw results of 10: Total generated: 63268, Percentage results: 6.3268%, Percentage of this or above: 25.603%
Saving throw results of 11: Total generated: 68256, Percentage results: 6.8256%, Percentage of this or above: 19.2762%
Saving throw results of 12: Total generated: 9692, Percentage results: 0.9692%, Percentage of this or above: 12.4506%
Saving throw results of 13: Total generated: 14988, Percentage results: 1.4988%, Percentage of this or above: 11.4814%
Saving throw results of 14: Total generated: 9900, Percentage results: 0.99%, Percentage of this or above: 9.9826%
Saving throw results of 15: Total generated: 15204, Percentage results: 1.5204%, Percentage of this or above: 8.9926%
Saving throw results of 16: Total generated: 10231, Percentage results: 1.0231%, Percentage of this or above: 7.4722%
Saving throw results of 17: Total generated: 13945, Percentage results: 1.3945%, Percentage of this or above: 6.4491%
Saving throw results of 18: Total generated: 8884, Percentage results: 0.8884%, Percentage of this or above: 5.0546%
Saving throw results of 19: Total generated: 11172, Percentage results: 1.1172%, Percentage of this or above: 4.1662%
Saving throw results of 20: Total generated: 6097, Percentage results: 0.6097%, Percentage of this or above: 3.049%
Saving throw results of 21: Total generated: 6694, Percentage results: 0.6694%, Percentage of this or above: 2.4393%
Saving throw results of 22: Total generated: 2971, Percentage results: 0.2971%, Percentage of this or above: 1.7699%
Saving throw results of 23: Total generated: 3611, Percentage results: 0.3611%, Percentage of this or above: 1.4728%
Saving throw results of 24: Total generated: 1346, Percentage results: 0.1346%, Percentage of this or above: 1.1117%
Saving throw results of 25: Total generated: 1870, Percentage results: 0.187%, Percentage of this or above: 0.9771%
Saving throw results of 26: Total generated: 1079, Percentage results: 0.1079%, Percentage of this or above: 0.7901%
Saving throw results of 27: Total generated: 1420, Percentage results: 0.142%, Percentage of this or above: 0.6822%
Saving throw results of 28: Total generated: 806, Percentage results: 0.0806%, Percentage of this or above: 0.5402%
Saving throw results of 29: Total generated: 1131, Percentage results: 0.1131%, Percentage of this or above: 0.4596%
Saving throw results of 30: Total generated: 633, Percentage results: 0.0633%, Percentage of this or above: 0.3465%
Saving throw results of 31: Total generated: 735, Percentage results: 0.0735%, Percentage of this or above: 0.2832%
Saving throw results of 32: Total generated: 379, Percentage results: 0.0379%, Percentage of this or above: 0.2097%
Saving throw results of 33: Total generated: 431, Percentage results: 0.0431%, Percentage of this or above: 0.1718%
Saving throw results of 34: Total generated: 208, Percentage results: 0.0208%, Percentage of this or above: 0.1287%
Saving throw results of 35: Total generated: 270, Percentage results: 0.027%, Percentage of this or above: 0.1079%
Saving throw results of 36: Total generated: 124, Percentage results: 0.0124%, Percentage of this or above: 0.0809%
Saving throw results of 37: Total generated: 154, Percentage results: 0.0154%, Percentage of this or above: 0.0685%
Saving throw results of 38: Total generated: 75, Percentage results: 0.0075%, Percentage of this or above: 0.0531%
Saving throw results of 39: Total generated: 121, Percentage results: 0.0121%, Percentage of this or above: 0.0456%
Saving throw results of 40: Total generated: 52, Percentage results: 0.0052%, Percentage of this or above: 0.0335%
Saving throw results of 41: Total generated: 67, Percentage results: 0.0067%, Percentage of this or above: 0.0283%
Saving throw results of 42: Total generated: 39, Percentage results: 0.0039%, Percentage of this or above: 0.0216%
Saving throw results of 43: Total generated: 38, Percentage results: 0.0038%, Percentage of this or above: 0.0177%
Saving throw results of 44: Total generated: 25, Percentage results: 0.0025%, Percentage of this or above: 0.0139%
Saving throw results of 45: Total generated: 27, Percentage results: 0.0027%, Percentage of this or above: 0.0114%
Saving throw results of 46: Total generated: 16, Percentage results: 0.0016%, Percentage of this or above: 0.0087%
Saving throw results of 47: Total generated: 13, Percentage results: 0.0013%, Percentage of this or above: 0.0071%
Saving throw results of 48: Total generated: 8, Percentage results: 0.0008%, Percentage of this or above: 0.0058%
Saving throw results of 49: Total generated: 9, Percentage results: 0.0009%, Percentage of this or above: 0.005%
Saving throw results of 50: Total generated: 12, Percentage results: 0.0012%, Percentage of this or above: 0.0041%
Saving throw results of 51: Total generated: 9, Percentage results: 0.0009%, Percentage of this or above: 0.0029%
Saving throw results of 52: Total generated: 5, Percentage results: 0.0005%, Percentage of this or above: 0.002%
Saving throw results of 53: Total generated: 6, Percentage results: 0.0006%, Percentage of this or above: 0.0015%
Saving throw results of 54: Total generated: 2, Percentage results: 0.0002%, Percentage of this or above: 0.0009%
Saving throw results of 55: Total generated: 2, Percentage results: 0.0002%, Percentage of this or above: 0.0007%
Saving throw results of 56: Total generated: , Percentage results: 0%, Percentage of this or above: 0.0005%
Saving throw results of 57: Total generated: 2, Percentage results: 0.0002%, Percentage of this or above: 0.0005%
Saving throw results of 58: Total generated: 1, Percentage results: 0.0001%, Percentage of this or above: 0.0003%
Saving throw results of 59: Total generated: , Percentage results: 0%, Percentage of this or above: 0.0002%
Saving throw results of 60: Total generated: 2, Percentage results: 0.0002%, Percentage of this or above: 0.0002%

Take-aways:

If you need a roll of 10 or more (say, what a Luck 10 character needs to beat a L1SR-LUK, or a DEX 15 character needs to beat a L2SR-DEX) you have about a 25.6% chance of making it.

If you need a roll of 5 or more (say, DEX 15 character needs to beat a L1SR-DEX) they have a 88.9% chance.

Need a roll of 15 or more: 8.9%

Need a roll of 20 or more: 3.04%

Need a roll of 25 or more: 0.977%

The closest thing to a 50/50 is a roll of 8 or more, 49.2%

Stefan

[mgtremaine]

You had me at little perl script

-Mike

[stefanj]

Lots of debug print statements in there you should remove, but here it is:

#!/usr/bin/perl

# DARO throw Monte Carlo simulator

# d6 thrower


# DARO thrower
sub daro_throw()
{
$done = 0;
$running_total=0;
while ($done !=1)
{
$die_one = int(rand 6)+1;
$die_two = int(rand 6)+1;
$running_total+=($die_one + $die_two);
print "Die 1: $die_one, Die 2: $die_two Running total: $running_total \n";
if ($die_one!=$die_two)
{
$done=1;
}
else
{
print "Doubles!\n";
}
}
return($running_total)
}

# Variables and arrays
$trials = 1000000;
@results = ();
@percentages = ();
@cumulative = ();
@total = ();

# Main loop

# Generate "$trials" die rolls and record the results
print @total;

for ($test=1;$test <= $trials;$test++)
{

$results[$test] = daro_throw();
print "Trial: $test Total: $results[$test]\n";
}

#print @total;

# Populate array
for ($result=3;$result < 61;$result++)
{
for ($test=1;$test <= $trials;$test++)
{

if ($results[$test] == $result)
{
$total[$result]++;
}
}
print "Save Roll total of $result found $total[$result] times\n";
}


# Find percentages
for ($result=3;$result < 61;$result++)
{
$percentages[$result]=($total[$result]/$trials)*100;
print "$percentages[$result]\n";
}

# Find @cumulative chance
for ($result=3;$result < 61;$result++)
{
for($above=$result;$above<61;$above++)
{
$cumulative[$result]+=$percentages[$above];
}
}

# Display results
for($result=3;$result<61;$result++)
{

print "Saving throw results of $result: Total generated: $total[$result], Percentage results: $percentages[$result]%, Percentage of this or above: $cumulative[$result]%\n";
}

#END

[craulabesh]

By the way: for each additional 5 points to beat, the success probability decreases to about 1/3.