Maths guys - hey! (Probabilities)

[Morrus]

That does highlight a flaw in the mechanic - the numbers are actually quite high. On a 5d6 or 6d6 (which wouldn't be uncommon) the duration tends to be much longer than a combat would be expected to take. Darnit. I suppose I could make it 5 or 6, though each little adjustments takes it a tiny step away from the elegance of the base concept!

 

[GMMichael]

That does highlight a flaw in the mechanic - the numbers are actually quite high. On a 5d6 or 6d6 (which wouldn't be uncommon) the duration tends to be much longer than a combat would be expected to take. Darnit. I suppose I could make it 5 or 6, though each little adjustments takes it a tiny step away from the elegance of the base concept!

For time limits, feel free to co-opt a rule from Modos RPG that I co-opted from another thread guru: everything has hit points.

For a death time limit, you'd set the clock at the character's full hit points, or maybe his health/stamina score. The speed of the clock is the size (or number, I guess) of die you use. Each round, GM or player rolls the die to see how much the hands more toward the end of the clock.

So the end is known. How fast you get there isn't.

 

[Truename]

Okay, I've written a Monte-Carlo simulation to answer these questions.

For a countdown pool where you remove dice showing a 6:

Pool Size: Median Rounds / Mode (Average)

1: <4 / 1 (6.0)
2: <7 / 4 (8.7)
3: <9 / 7 (10.6)
4: =10 / 9 (11.9)
5: >11 / 9 (13.0)
6: >12 / 10 (13.9)

Here's the histograms:

Simulating countdown pool (6 or higher) with 10000 simulations

1d6 countdown (6 or higher)
  1 (16.5%): ===================================================================
  2 (30.4%): ========================================================
  3 (42.0%): ==============================================
  4 (51.6%): ======================================
  5 (59.9%): =================================
  6 (66.5%): ===========================
  7 (72.2%): ======================
  8 (77.2%): ====================
  9 (80.9%): ===============
 10 (84.0%): ============
 11 (86.8%): ===========
 12 (89.0%): ========
 13 (90.6%): ======
 14 (92.0%): ======
 15 (93.4%): =====
 16 (94.6%): ====
 17 (95.5%): ===
 18 (96.2%): ===
 19 (96.9%): ==
 20 (97.3%): =
 21 (97.8%): =
 22 (98.2%): =
 24 (98.7%): =

2d6 countdown (6 or higher)
  1  (2.6%): ==================
  2  (9.0%): ===============================================
  3 (17.4%): =============================================================
  4 (26.5%): ===================================================================
  5 (35.4%): =================================================================
  6 (44.0%): ===============================================================
  7 (51.6%): =======================================================
  8 (58.7%): ====================================================
  9 (64.9%): =============================================
 10 (70.2%): =======================================
 11 (74.7%): ================================
 12 (78.5%): ============================
 13 (81.9%): ========================
 14 (85.1%): =======================
 15 (87.5%): =================
 16 (89.9%): =================
 17 (91.5%): ===========
 18 (92.8%): ========
 19 (93.9%): ========
 20 (95.0%): ========
 21 (95.9%): ======
 22 (96.8%): ======
 23 (97.4%): ===
 24 (97.7%): ==
 25 (98.1%): ==
 26 (98.5%): ==
 27 (98.7%): =
 28 (98.9%): =
 30 (99.2%): =

3d6 countdown (6 or higher)
  1  (0.5%): ====
  2  (2.9%): ==================
  3  (7.6%): ====================================
  4 (13.8%): ================================================
  5 (21.1%): ========================================================
  6 (28.6%): ==========================================================
  7 (37.2%): ===================================================================
  8 (44.5%): ========================================================
  9 (51.6%): =======================================================
 10 (58.3%): ===================================================
 11 (63.9%): ===========================================
 12 (69.2%): =========================================
 13 (73.9%): ====================================
 14 (78.1%): ================================
 15 (81.5%): ==========================
 16 (84.5%): ======================
 17 (87.1%): ====================
 18 (89.2%): ================
 19 (91.2%): ===============
 20 (92.6%): ==========
 21 (93.9%): ==========
 22 (94.8%): =======
 23 (95.6%): ======
 24 (96.3%): =====
 25 (97.0%): ====
 26 (97.6%): ====
 27 (98.0%): ==
 28 (98.3%): ==
 29 (98.6%): ==
 30 (98.9%): ==
 31 (99.2%): ==
 32 (99.3%): =
 33 (99.5%): =

4d6 countdown (6 or higher)
  2  (0.9%): =======
  3  (3.3%): ====================
  4  (7.3%): =================================
  5 (12.9%): ================================================
  6 (19.9%): ==========================================================
  7 (27.5%): ================================================================
  8 (35.0%): ===============================================================
  9 (42.9%): ===================================================================
 10 (50.3%): ==============================================================
 11 (56.8%): =======================================================
 12 (62.5%): ================================================
 13 (68.0%): ==============================================
 14 (73.0%): ==========================================
 15 (76.8%): ===============================
 16 (80.2%): =============================
 17 (83.2%): =========================
 18 (85.8%): ======================
 19 (88.0%): ==================
 20 (90.0%): ================
 21 (91.8%): ==============
 22 (92.9%): =========
 23 (93.9%): ========
 24 (95.0%): ========
 25 (95.8%): =======
 26 (96.5%): ======
 27 (97.0%): ====
 28 (97.5%): ===
 29 (97.9%): ===
 30 (98.3%): ===
 31 (98.7%): ==
 32 (98.9%): =
 33 (99.2%): ==
 34 (99.3%): =
 35 (99.5%): =

5d6 countdown (6 or higher)
  2  (0.2%): =
  3  (1.2%): ========
  4  (3.5%): =====================
  5  (7.4%): ==================================
  6 (12.9%): ================================================
  7 (19.5%): ==========================================================
  8 (26.6%): ==============================================================
  9 (34.1%): ===================================================================
 10 (41.5%): =================================================================
 11 (48.7%): ================================================================
 12 (55.6%): =============================================================
 13 (61.5%): ====================================================
 14 (67.1%): ==================================================
 15 (71.9%): ==========================================
 16 (76.3%): ======================================
 17 (79.7%): ==============================
 18 (82.7%): ==========================
 19 (85.4%): =======================
 20 (87.6%): ====================
 21 (89.7%): ==================
 22 (91.2%): =============
 23 (92.7%): =============
 24 (93.8%): =========
 25 (95.0%): ==========
 26 (95.8%): ======
 27 (96.5%): ======
 28 (97.0%): =====
 29 (97.4%): ===
 30 (97.9%): ====
 31 (98.2%): ==
 32 (98.4%): ==
 33 (98.7%): ==
 34 (98.9%): =
 35 (99.1%): ==
 36 (99.3%): =

6d6 countdown (6 or higher)
  3  (0.6%): ====
  4  (1.9%): ===========
  5  (4.6%): =======================
  6  (8.8%): =====================================
  7 (14.0%): ==============================================
  8 (20.5%): =========================================================
  9 (27.3%): ============================================================
 10 (34.9%): ===================================================================
 11 (42.1%): ================================================================
 12 (49.5%): =================================================================
 13 (55.6%): =====================================================
 14 (61.6%): ====================================================
 15 (67.2%): =================================================
 16 (71.9%): =========================================
 17 (76.1%): =====================================
 18 (79.3%): ============================
 19 (82.7%): ==============================
 20 (85.7%): =========================
 21 (87.9%): ====================
 22 (90.0%): =================
 23 (91.5%): =============
 24 (92.8%): ===========
 25 (93.7%): ========
 26 (94.7%): ========
 27 (95.5%): =======
 28 (96.2%): ======
 29 (96.8%): =====
 30 (97.4%): =====
 31 (97.9%): ====
 32 (98.4%): ===
 33 (98.7%): ===
 34 (98.9%): =
 35 (99.1%): =
 36 (99.2%): =
 37 (99.4%): =

Averages:
1: 5.9995
2: 8.7283
3: 10.5784
4: 11.8578
5: 13.0125
6: 13.9011

If you remove dice showing a 5 or a 6:

1: <2 / 1 (3.0)
2: =3 / 2 (4.2)
3: <4 / 3 (5.0)
4: 4-5 / 4 (5.6)
5: =5 / 4 (6.1)
6: 5-6 / 5 (6.6)

Histograms:

Simulating countdown pool (5 or higher) with 10000 simulations

1d6 countdown (5 or higher)
  1 (33.3%): ===================================================================
  2 (56.0%): =============================================
  3 (70.6%): =============================
  4 (80.4%): ===================
  5 (86.8%): ============
  6 (91.2%): ========
  7 (94.0%): =====
  8 (96.0%): ===
  9 (97.3%): ==
 10 (98.2%): =
 11 (98.8%): =

2d6 countdown (5 or higher)
  1 (11.1%): ======================================
  2 (30.4%): ===================================================================
  3 (49.4%): ==================================================================
  4 (64.5%): ====================================================
  5 (75.6%): ======================================
  6 (83.5%): ===========================
  7 (88.8%): ==================
  8 (92.5%): ============
  9 (94.8%): =======
 10 (96.4%): =====
 11 (97.5%): ===
 12 (98.4%): ==
 13 (98.9%): =
 14 (99.4%): =

3d6 countdown (5 or higher)
  1  (3.7%): =============
  2 (17.2%): ==================================================
  3 (35.3%): ===================================================================
  4 (52.6%): ================================================================
  5 (66.0%): =================================================
  6 (76.1%): =====================================
  7 (83.8%): ============================
  8 (89.0%): ===================
  9 (92.3%): ============
 10 (95.0%): =========
 11 (96.6%): ======
 12 (97.8%): ====
 13 (98.6%): ===
 14 (99.1%): =

4d6 countdown (5 or higher)
  1  (1.2%): ====
  2  (9.7%): ==================================
  3 (25.1%): ==============================================================
  4 (41.7%): ===================================================================
  5 (57.4%): ===============================================================
  6 (69.1%): ===============================================
  7 (78.6%): ======================================
  8 (85.2%): ==========================
  9 (90.2%): ====================
 10 (93.3%): ============
 11 (95.5%): ========
 12 (97.0%): ======
 13 (98.1%): ====
 14 (98.8%): ==
 15 (99.2%): =
 16 (99.5%): =

5d6 countdown (5 or higher)
  1  (0.5%): ==
  2  (5.1%): ==================
  3 (17.6%): ===================================================
  4 (33.8%): ===================================================================
  5 (50.0%): ==================================================================
  6 (63.5%): =======================================================
  7 (73.8%): ==========================================
  8 (81.8%): =================================
  9 (87.4%): =======================
 10 (91.6%): =================
 11 (94.4%): ===========
 12 (96.2%): =======
 13 (97.4%): ====
 14 (98.2%): ===
 15 (98.8%): ==
 16 (99.3%): =

6d6 countdown (5 or higher)
  2  (2.8%): ==========
  3 (11.6%): ===================================
  4 (26.0%): =========================================================
  5 (42.6%): ===================================================================
  6 (57.4%): ===========================================================
  7 (69.5%): ================================================
  8 (78.6%): ====================================
  9 (85.2%): ==========================
 10 (90.0%): ===================
 11 (93.3%): =============
 12 (95.4%): ========
 13 (96.9%): =====
 14 (97.9%): ====
 15 (98.6%): ==
 16 (99.1%): =
 17 (99.4%): =

Averages:
1: 2.9958
2: 4.2037
3: 4.9902
4: 5.616
5: 6.1191
6: 6.5664

If you remove dice showing 4-6:

1: =1 / 1 (2.0)
2: <2 / 2 (2.9)
3: >2 / 2 (3.1)
4: <3 / 3 (3.5)
5: =3 / 3 (3.7)
6: >3 / 3 (4.0)

Histograms:

Simulating countdown pool (4 or higher) with 10000 simulations

1d6 countdown (4 or higher)
  1 (49.0%): ===================================================================
  2 (75.0%): ===================================
  3 (87.9%): =================
  4 (93.7%): =======
  5 (96.8%): ====
  6 (98.4%): ==
  7 (99.3%): =

2d6 countdown (4 or higher)
  1 (24.8%): =====================================================
  2 (56.2%): ===================================================================
  3 (76.2%): ==========================================
  4 (87.4%): =======================
  5 (93.4%): ============
  6 (96.7%): =======
  7 (98.5%): ===
  8 (99.2%): =

3d6 countdown (4 or higher)
  1 (12.7%): ============================
  2 (42.1%): ===================================================================
  3 (67.6%): =========================================================
  4 (82.3%): =================================
  5 (90.9%): ===================
  6 (95.6%): ==========
  7 (97.8%): =====
  8 (98.9%): ==
  9 (99.5%): =

4d6 countdown (4 or higher)
  1  (6.2%): ===============
  2 (32.1%): ==================================================================
  3 (58.2%): ===================================================================
  4 (76.7%): ===============================================
  5 (88.2%): =============================
  6 (93.8%): ==============
  7 (96.8%): =======
  8 (98.3%): ===
  9 (99.1%): =
 10 (99.6%): =

5d6 countdown (4 or higher)
  1  (3.3%): =======
  2 (23.9%): ==================================================
  3 (51.2%): ===================================================================
  4 (73.0%): =====================================================
  5 (85.9%): ===============================
  6 (92.9%): =================
  7 (96.2%): ========
  8 (98.0%): ====
  9 (99.0%): ==
 10 (99.5%): =

6d6 countdown (4 or higher)
  1  (1.4%): ===
  2 (18.2%): =========================================
  3 (45.2%): ===================================================================
  4 (67.6%): =======================================================
  5 (82.8%): =====================================
  6 (90.9%): ====================
  7 (95.4%): ===========
  8 (97.7%): =====
  9 (98.8%): ==
 10 (99.3%): =

Averages:
1: 2.0057
2: 2.6874
3: 3.1319
4: 3.5146
5: 3.7781
6: 4.0316

And here's the code I used (not including supporting libraries). It's Ruby.

require 'histogram'
require 'dice'
require 'report'

$NUM_SIMS = 10000
$COUNTDOWN_DIE = Integer(ARGV[0])

raise "Usage: main.rb [countdown die]" if $COUNTDOWN_DIE <= 0

def roll_pool(pool_size)
      pool = pool_size.d6
      num_sixes = pool.rolls.select { |e| e >= $COUNTDOWN_DIE }.length
#      puts pool.details
#      puts "# of 6s: #{num_sixes}"
      return num_sixes
end

def simulate_pool(pool_size)
    rounds = 0
  while pool_size > 0 do
      pool_size -= roll_pool(pool_size)
      rounds += 1
  end
  return rounds
end

def run_sim(pool_size)
    puts "#{pool_size}d6 countdown (#{$COUNTDOWN_DIE} or higher)"
    rounds = Histogram.new
    $NUM_SIMS.times do
        rounds.increment(simulate_pool pool_size)
    end
    rounds.print(STDOUT)
    return rounds.average
end

puts "Simulating countdown pool with #{$NUM_SIMS} simulations"

average = []
1.upto 6 do |pool_size|
    puts
  average[pool_size] = run_sim pool_size
end

puts
puts "Averages:"
1.upto 6 do |pool_size|
  puts "#{pool_size}: #{average[pool_size]}"
end

 

[Morrus]

Oh, wow - that's really useful! Thank you!

 

[Tom Strickland]

Just to add a reference because of the topic subject matter:

I recalled this thread when I saw this post at Java-Gaming.org:

Using random number generators

Which points out this Jan 2012 article by Red Blob Games:

Probability and Games: Damage Rolls

 

[doghead]

For time limits, feel free to co-opt a rule from Modos RPG that I co-opted from another thread guru: everything has hit points.

For a death time limit, you'd set the clock at the character's full hit points, or maybe his health/stamina score. The speed of the clock is the size (or number, I guess) of die you use. Each round, GM or player rolls the die to see how much the hands more toward the end of the clock.

So the end is known. How fast you get there isn't.

There are a number of interesting differences between this system and the one proposed. In this system, there are more hard certainties. Consider the following example, Hit Points: 12, Depleting die: 1d6, rolls occur at the end of the turn.

You know the following: You have a minimum of 2 turns before reaching 0 (two rolls of 6). You have a maximum of 12 turns (12 rolls of 1). If your hit points reach 1, you have one turn remaining.

In the proposed system it is possible to roll all 6's in the first turn. Also, when reduced to 1d6, you may survive one turn, or six, or even more.

Which you prefer would depend upon what you are trying to achieve and your playing style.

thotd

 

[doghead]

I've co-opted it for death saves and for suffocation. It works great there

Are there any mechanisms (apart from restocking) for inserting other variables into the mix. For example, the ability of a "tough" character to resist disease or poison more effectively than a "frail" character.

Actually, thats not a great example as you are using Endurance as a starting point, so its already factored in. How about a mechanism for differentiating "strong" diseases vs "weak" diseases? I suppose that changing it from "remove 6's" to "remove 5's and 6's" is one option.

Another option might be to allow the use a form bonus and penalty points. A bonus of +1 allows the result of one dice to be increased by one. For example (assuming a "remove 6's setup) if a 5 is rolled, it could be increased to 6 and the dice removed. If the highest result is a 4, then the +1 bonus will have no effect, only allowing the result to be increased to a 5. God knows how this affects the probabilities (a good mathematician might as well).

So in the example of the guards searching for the character hiding in the crate, you might say that each additional guard involved in the search provides a +1 bonus. This allows for smaller incremental changes than say, moving from "remove 6's" to "remove 5's and 6's".

thotd