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Plinko

Last updated: June 04, 2020

Plinko

Introduction

Plinko is the general term for a game involving a ball or disk that falls randomly down a grid of pegs, known as a Galton Board, to land in a random spot at the bottom. Such boards are also a good way to illustrate a Gaussian curve at science museums. In the case of the casino game, the concept is the same, but where the ball lands will determine how much the player wins/loses.

This page presents three variants of Plinko I have seen, by CryptoGames, BGAMING, and BetFury.

Provably Fair Casinos

Name		Wizards Seal	US-OH Friendly	Play
	CryptoSlots Casino 3.8	Yes	Yes	Visit Casino

Name:

CryptoSlots Casino

3.8

Wizards Seal: Yes

US-OH Friendly: Yes

Play: Visit Casino

CryptoGames Rules

Following are the rules for how the game is played at CryptoGames.

The player chooses a bet and a pay table (green, red, blue, or yellow).
Bets are made in "credits." A credit costs 0.000001 BTC (Bitcoin) and the minimum bet is 10 credits.
A ball is dropped from above a triangular field of pegs, with 16 rows and 17 possible places to land at the bottom.
At each row, the ball will hit a peg and may either go left or right, each with a 50% chance.
When the ball reaches one of the 17 possible resting spots at the bottom, the player will be paid according to amount bet, where the ball landed, and pay table chosen.

Per the concept of "provably fair gaming" common to Cryto-casinos, the final outcome is determined in advance. With 16 rows of pegs, there are 2^16 = 65,536 possible paths the ball can take. The final outcome is based on a random number from 0 to 65,535, which gets mapped according to one of the positions at the bottom of the grid. I have checked and the mapping is done correctly, according to the natural odds of a fair Plinko board.

The following table shows the number of combinations for each outcome, the range associated with it, and what each of the four pay tables pay for that position. All pays are on a "for one" basis.

Plinko Pay Table

Position	Combinations	Range	Green Pays	Red Pays	Blue Pays	Yellow Pays
0	1	0 to 0	10	20	50	650
1	16	1 to 16	8	7	8	30
2	120	17 to 136	6	5	3	7
3	560	137 to 696	3	3	2	3
4	1,820	697 to 2516	2	2	1.4	1.5
5	4,368	2517 to 6884	1.3	1.1	1.2	1.2
6	8,008	6885 to 14892	1	1	1.1	1
7	11,440	14893 to 26332	0.8	0.6	1	0.7
8	12,870	26333 to 39202	0.5	1	0.4	0.7
9	11,440	39203 to 50642	0.8	0.6	1	0.7
10	8,008	50643 to 58650	1	1	1.1	1
11	4,368	58651 to 63018	1.3	1.1	1.2	1.2
12	1,820	63019 to 64838	2	2	1.4	1.5
13	560	64839 to 65398	3	3	2	3
14	120	65399 to 65518	6	5	3	7
15	16	65519 to 65534	8	7	8	30
16	1	65535 to 65535	10	20	50	650

Analysis

The following table shows my analysis of the Green pay table. The lower right cell shows an expected return of 98.37%.

Green Pay Table Analysis

Position	Combinations	Pays	Probability	Return
0	1	10	0.000015	0.000153
1	16	8	0.000244	0.001953
2	120	6	0.001831	0.010986
3	560	3	0.008545	0.025635
4	1,820	2	0.027771	0.055542
5	4,368	1.3	0.066650	0.086646
6	8,008	1	0.122192	0.122192
7	11,440	0.8	0.174561	0.139648
8	12,870	0.5	0.196381	0.098190
9	11,440	0.8	0.174561	0.139648
10	8,008	1	0.122192	0.122192
11	4,368	1.3	0.066650	0.086646
12	1,820	2	0.027771	0.055542
13	560	3	0.008545	0.025635
14	120	6	0.001831	0.010986
15	16	8	0.000244	0.001953
16	1	10	0.000015	0.000153
Total	65,536		1.000000	0.983701

The following table shows my analysis of the Red pay table. The lower right cell shows an expected return of 98.16%.

Red Pay Table Analysis

Position	Combinations	Pays	Probability	Return
0	1	20	0.000015	0.000305
1	16	7	0.000244	0.001709
2	120	5	0.001831	0.009155
3	560	3	0.008545	0.025635
4	1,820	2	0.027771	0.055542
5	4,368	1.1	0.066650	0.073315
6	8,008	1	0.122192	0.122192
7	11,440	0.6	0.174561	0.104736
8	12,870	1	0.196381	0.196381
9	11,440	0.6	0.174561	0.104736
10	8,008	1	0.122192	0.122192
11	4,368	1.1	0.066650	0.073315
12	1,820	2	0.027771	0.055542
13	560	3	0.008545	0.025635
14	120	5	0.001831	0.009155
15	16	7	0.000244	0.001709
16	1	20	0.000015	0.000305
Total	65,536		1.000000	0.981561

The following table shows my analysis of the Blue pay table. The lower right cell shows an expected return of 98.37%.

Blue Pay Table Analysis

Position	Combinations	Pays	Probability	Return
0	1	50	0.000015	0.000763
1	16	8	0.000244	0.001953
2	120	3	0.001831	0.005493
3	560	2	0.008545	0.017090
4	1,820	1.4	0.027771	0.038879
5	4,368	1.2	0.066650	0.079980
6	8,008	1.1	0.122192	0.134412
7	11,440	1	0.174561	0.174561
8	12,870	0.4	0.196381	0.078552
9	11,440	1	0.174561	0.174561
10	8,008	1.1	0.122192	0.134412
11	4,368	1.2	0.066650	0.079980
12	1,820	1.4	0.027771	0.038879
13	560	2	0.008545	0.017090
14	120	3	0.001831	0.005493
15	16	8	0.000244	0.001953
16	1	50	0.000015	0.000763
Total	65,536		1.000000	0.984814

The following table shows my analysis of the Yellow pay table. The lower right cell shows an expected return of 98.09%.

Yellow Pay Table Analysis

Position	Combinations	Pays	Probability	Return
0	1	650	0.000015	0.009918
1	16	30	0.000244	0.007324
2	120	7	0.001831	0.012817
3	560	3	0.008545	0.025635
4	1,820	1.5	0.027771	0.041656
5	4,368	1.2	0.066650	0.079980
6	8,008	1	0.122192	0.122192
7	11,440	0.7	0.174561	0.122192
8	12,870	0.7	0.196381	0.137466
9	11,440	0.7	0.174561	0.122192
10	8,008	1	0.122192	0.122192
11	4,368	1.2	0.066650	0.079980
12	1,820	1.5	0.027771	0.041656
13	560	3	0.008545	0.025635
14	120	7	0.001831	0.012817
15	16	30	0.000244	0.007324
16	1	650	0.000015	0.009918
Total	65,536		1.000000	0.980899

Summary

The following table shows the return and standard deviation for each pay table. The greatest return is for the Blue pay table, at 98.48%.

Summary

Pay Table	Return	Standard Deviation
Green	98.37%	0.562711
Red	98.16%	0.517632
Blue	98.48%	0.464829
Yellow	98.09%	3.678698

Probably Fair

The way "provably fair" mechanism works, at least at Crypto.Games, is as follows:

Enter a string consisting of the casino hash followed by the player hash into a SHA512 hash generator, like this one.
Convert the first four characters in the hash from step 1 into a hexi-decimal to base-10 converter, like this one.
The result of step 2 will be an integer from 0 to 65535 (16⁴-1). This result is mapped to an outcome according to the ranges in the first table of this page.

The way this is "provably fair," is the casino will provide the hash of its own seed before you bet, ensuring the casino's contribution to the outcome was predestined, assuming the casino seed provided after the bet matches that hash.

If this sounds Greek to you, I go through the concepts and terminology of using cryptography for fair gaming in my page on Dice (Encrypted Version) more slowly.

If you want to simplify the process above, I welcome you to use this program I saved at PHP Sandbox. Just enter the Server Seed on line 4, the Client Seed on line 5, and click "execute code." The game outcome will be in the Results box below the code.

You may also see my code by clicking the button below.

// Plinko game conversion for Crypto.Games

$server_seed = "k34pQFblHvQAJ33zZZHCQtlFlhHb4KTtw2qhOahC";
$client_seed = "6VoqO9MXdSp5xSmiq2L6xcvn2XVvFWVLkC0TtLwc";
$combined_seed = $server_seed.$client_seed;
echo "Combined seed = $combined_seed\n";
$combined_hash = hash('sha512', $combined_seed);
echo "Hash of combined seed = $combined_hash\n";
$first_four=substr($combined_hash,0,4);
echo "First four characters = $first_four\n";
$hex_to_dec=hexdec($first_four);
echo "Converted to decimal = $hex_to_dec\n";
$weight_array=array(1,17,137,697,2517,6885,14893,26333,39203,50643,58651,63019,64839,65399,65519,65535,65536);
$green_array=array(10,8,6,3,2,1.3,1,0.8,0.5,0.8,1,1.3,2,3,6,8,10);
$red_array=array(20,7,5,3,2,1.1,1,0.6,1,0.6,1,1.1,2,3,5,7,20);
$blue_array=array(50,8,3,2,1.4,1.2,1.1,1,0.4,1,1.1,1.2,1.4,2,3,8,50);
$yellow_array=array(650,30,7,3,1.5,1.2,1,0.7,0.7,0.7,1,1.2,1.5,3,7,30,650);
$i=0;
while ( $hex_to_dec >= $weight_array[$i] ) {
$i++;
}
echo "Green win = \t$green_array[$i]\n";
echo "Red win = \t$red_array[$i]\n";
echo "Blue win = \t$blue_array[$i]\n";
echo "Yellow win = \t$yellow_array[$i]\n";

// Procedure
// 1. Join server and client seeds, server seed first.
// 2. Generate a SHA-512 hash of the string from step 1.
// 3. Convert first FOUR characters of the hash from hexidecimal to decimal.
// 4. Convert result from step 3 to a win per the arrays given.
?>

BGAMING Rules

BGAMING follows the same concept of balls dropping down a Galton Board. The player may choose from 8 to 16 rows as well as low, medium, or high volatility.

To keep this page from running too long, I present one table for each number of rows, showing the pay tables and return for all three pay tables. The Position column shows how many positions away from the left-most position the ball lands.

16 Rows

Low Risk Pays	Medium Risk Pays	High Risk Pays	Combinations	Probability	Low Risk Return	Medium Risk Return	High Risk Return
0	16	110	1000	1	0.000015	0.001678	0.001678	0.015259
1	9	41	130	16	0.000244	0.010010	0.010010	0.031738
2	2	10	26	120	0.001831	0.018311	0.018311	0.047607
3	1.4	5	9	560	0.008545	0.042725	0.042725	0.076904
4	1.4	3	4	1,820	0.027771	0.083313	0.083313	0.111084
5	1.2	1.5	2	4,368	0.066650	0.099976	0.099976	0.133301
6	1.1	1	0.2	8,008	0.122192	0.122192	0.122192	0.024438
7	1	0.5	0.2	11,440	0.174561	0.087280	0.087280	0.034912
8	0.5	0.3	0.2	12,870	0.196381	0.058914	0.058914	0.039276
9	1	0.5	0.2	11,440	0.174561	0.087280	0.087280	0.034912
10	1.1	1	0.2	8,008	0.122192	0.122192	0.122192	0.024438
11	1.2	1.5	2	4,368	0.066650	0.099976	0.099976	0.133301
12	1.4	3	4	1,820	0.027771	0.083313	0.083313	0.111084
13	1.4	5	9	560	0.008545	0.042725	0.042725	0.076904
14	2	10	26	120	0.001831	0.018311	0.018311	0.047607
15	9	41	130	16	0.000244	0.010010	0.010010	0.031738
16	16	110	1000	1	0.000015	0.001678	0.001678	0.015259
Total				65,536	1.000000	0.989883	0.989883	0.989764

15 Rows

Position	Low Risk Pays	Medium Risk Pays	High Risk Pays	Combinations	Probability	Low Risk Return	Medium Risk Return	High Risk Return
0	15	88	620	1	0.000031	0.002686	0.002686	0.018921
1	8	18	83	15	0.000458	0.008240	0.008240	0.037994
2	3	11	27	105	0.003204	0.035248	0.035248	0.086517
3	2	5	8	455	0.013885	0.069427	0.069427	0.111084
4	1.5	3	3	1,365	0.041656	0.124969	0.124969	0.124969
5	1.1	1.3	0.5	3,003	0.091644	0.119138	0.119138	0.045822
6	1	0.5	0.2	5,005	0.152740	0.076370	0.076370	0.030548
7	0.7	0.3	0.2	6,435	0.196381	0.058914	0.058914	0.039276
8	0.7	0.3	0.2	6,435	0.196381	0.058914	0.058914	0.039276
9	1	0.5	0.2	5,005	0.152740	0.076370	0.076370	0.030548
10	1.1	1.3	0.5	3,003	0.091644	0.119138	0.119138	0.045822
11	1.5	3	3	1,365	0.041656	0.124969	0.124969	0.124969
12	2	5	8	455	0.013885	0.069427	0.069427	0.111084
13	3	11	27	105	0.003204	0.035248	0.035248	0.086517
14	8	18	83	15	0.000458	0.008240	0.008240	0.037994
15	15	88	620	1	0.000031	0.002686	0.002686	0.018921
Total				32,768	1.000000	0.989984	0.989984	0.990265

14 Rows

Position	Low Risk Pays	Medium Risk Pays	High Risk Pays	Combinations	Probability	Low Risk Return	Medium Risk Return	High Risk Return
0	7.1	58	420	1	0.000061	0.003540	0.003540	0.025635
1	4	15	56	14	0.000854	0.012817	0.012817	0.047852
2	1.9	7	18	91	0.005554	0.038879	0.038879	0.099976
3	1.4	4	5	364	0.022217	0.088867	0.088867	0.111084
4	1.3	1.9	1.9	1,001	0.061096	0.116083	0.116083	0.116083
5	1.1	1	0.3	2,002	0.122192	0.122192	0.122192	0.036658
6	1	0.5	0.2	3,003	0.183289	0.091644	0.091644	0.036658
7	0.5	0.2	0.2	3,432	0.209473	0.041895	0.041895	0.041895
8	1	0.5	0.2	3,003	0.183289	0.091644	0.091644	0.036658
9	1.1	1	0.3	2,002	0.122192	0.122192	0.122192	0.036658
10	1.3	1.9	1.9	1,001	0.061096	0.116083	0.116083	0.116083
11	1.4	4	5	364	0.022217	0.088867	0.088867	0.111084
12	1.9	7	18	91	0.005554	0.038879	0.038879	0.099976
13	4	15	56	14	0.000854	0.012817	0.012817	0.047852
14	7.1	58	420	1	0.000061	0.003540	0.003540	0.025635
Total				16,384	1.000000	0.989941	0.989941	0.989783

13 Rows

Position	Low Risk Pays	Medium Risk Pays	High Risk Pays	Combinations	Probability	Low Risk Return	Medium Risk Return	High Risk Return
0	8.1	43	260	1	0.000122	0.005249	0.005249	0.031738
1	4	13	37	13	0.001587	0.020630	0.020630	0.058716
2	3	6	11	78	0.009521	0.057129	0.057129	0.104736
3	1.9	3	4	286	0.034912	0.104736	0.104736	0.139648
4	1.2	1.3	1	715	0.087280	0.113464	0.113464	0.087280
5	0.9	0.7	0.2	1,287	0.157104	0.109973	0.109973	0.031421
6	0.7	0.4	0.2	1,716	0.209473	0.083789	0.083789	0.041895
7	0.7	0.4	0.2	1,716	0.209473	0.083789	0.083789	0.041895
8	0.9	0.7	0.2	1,287	0.157104	0.109973	0.109973	0.031421
9	1.2	1.3	1	715	0.087280	0.113464	0.113464	0.087280
10	1.9	3	4	286	0.034912	0.104736	0.104736	0.139648
11	3	6	11	78	0.009521	0.057129	0.057129	0.104736
12	4	13	37	13	0.001587	0.020630	0.020630	0.058716
13	8.1	43	260	1	0.000122	0.005249	0.005249	0.031738
Total				8,192	1.000000	0.989941	0.989941	0.990869

12 Rows

Position	Low Risk Pays	Medium Risk Pays	High Risk Pays	Combinations	Probability	Low Risk Return	Medium Risk Return	High Risk Return
0	10	33	170	1	0.000244	0.008057	0.008057	0.041504
1	3	11	24	12	0.002930	0.032227	0.032227	0.070313
2	1.6	4	8.1	66	0.016113	0.064453	0.064453	0.130518
3	1.4	2	2	220	0.053711	0.107422	0.107422	0.107422
4	1.1	1.1	0.7	495	0.120850	0.132935	0.132935	0.084595
5	1	0.6	0.2	792	0.193359	0.116016	0.116016	0.038672
6	0.5	0.3	0.2	924	0.225586	0.067676	0.067676	0.045117
7	1	0.6	0.2	792	0.193359	0.116016	0.116016	0.038672
8	1.1	1.1	0.7	495	0.120850	0.132935	0.132935	0.084595
9	1.4	2	2	220	0.053711	0.107422	0.107422	0.107422
10	1.6	4	8.1	66	0.016113	0.064453	0.064453	0.130518
11	3	11	24	12	0.002930	0.032227	0.032227	0.070313
12	10	33	170	1	0.000244	0.008057	0.008057	0.041504
Total				4,096	1.000000	0.989893	0.989893	0.991162

11 Rows

Position	Low Risk Pays	Medium Risk Pays	High Risk Pays	Combinations	Probability	Low Risk Return	Medium Risk Return	High Risk Return
0	8.4	24	120	1	0.000488	0.011719	0.011719	0.058594
1	3	6	14	11	0.005371	0.032227	0.032227	0.075195
2	1.9	3	5.2	55	0.026855	0.080566	0.080566	0.139648
3	1.3	1.8	1.4	165	0.080566	0.145020	0.145020	0.112793
4	1	0.7	0.4	330	0.161133	0.112793	0.112793	0.064453
5	0.7	0.5	0.2	462	0.225586	0.112793	0.112793	0.045117
6	0.7	0.5	0.2	462	0.225586	0.112793	0.112793	0.045117
7	1	0.7	0.4	330	0.161133	0.112793	0.112793	0.064453
8	1.3	1.8	1.4	165	0.080566	0.145020	0.145020	0.112793
9	1.9	3	5.2	55	0.026855	0.080566	0.080566	0.139648
10	3	6	14	11	0.005371	0.032227	0.032227	0.075195
11	8.4	24	120	1	0.000488	0.011719	0.011719	0.058594
Total				2,048	1.000000	0.990234	0.990234	0.991602

10 Rows

Position	Low Risk Pays	Medium Risk Pays	High Risk Pays	Combinations	Probability	Low Risk Return	Medium Risk Return	High Risk Return
0	8.9	22	76	1	0.000977	0.021484	0.021484	0.074219
1	3	5	10	10	0.009766	0.048828	0.048828	0.097656
2	1.4	2	3	45	0.043945	0.087891	0.087891	0.131836
3	1.1	1.4	0.9	120	0.117188	0.164063	0.164063	0.105469
4	1	0.6	0.3	210	0.205078	0.123047	0.123047	0.061523
5	0.5	0.4	0.2	252	0.246094	0.098438	0.098438	0.049219
6	1	0.6	0.3	210	0.205078	0.123047	0.123047	0.061523
7	1.1	1.4	0.9	120	0.117188	0.164063	0.164063	0.105469
8	1.4	2	3	45	0.043945	0.087891	0.087891	0.131836
9	3	5	10	10	0.009766	0.048828	0.048828	0.097656
10	8.9	22	76	1	0.000977	0.021484	0.021484	0.074219
Total				1,024	1.000000	0.989063	0.989063	0.990625

9 Rows

Position	Low Risk Pays	Medium Risk Pays	High Risk Pays	Combinations	Probability	Low Risk Return	Medium Risk Return	High Risk Return
0	5.6	18	43	1	0.001953	0.035156	0.035156	0.083984
1	2	4	7	9	0.017578	0.070313	0.070313	0.123047
2	1.6	1.7	2	36	0.070313	0.119531	0.119531	0.140625
3	1	0.9	0.6	84	0.164063	0.147656	0.147656	0.098438
4	0.7	0.5	0.2	126	0.246094	0.123047	0.123047	0.049219
5	0.7	0.5	0.2	126	0.246094	0.123047	0.123047	0.049219
6	1	0.9	0.6	84	0.164063	0.147656	0.147656	0.098438
7	1.6	1.7	2	36	0.070313	0.119531	0.119531	0.140625
8	2	4	7	9	0.017578	0.070313	0.070313	0.123047
9	5.6	18	43	1	0.001953	0.035156	0.035156	0.083984
Total				512	1.000000	0.991406	0.991406	0.990625

8 Rows

Position	Low Risk Pays	Medium Risk Pays	High Risk Pays	Combinations	Probability	Low Risk Return	Medium Risk Return	High Risk Return
0	5.6	13	29	1	0.003906	0.050781	0.050781	0.113281
1	2.1	3	4	8	0.031250	0.093750	0.093750	0.125000
2	1.1	1.3	1.5	28	0.109375	0.142188	0.142188	0.164063
3	1	0.7	0.3	56	0.218750	0.153125	0.153125	0.065625
4	0.5	0.4	0.2	70	0.273438	0.109375	0.109375	0.054688
5	1	0.7	0.3	56	0.218750	0.153125	0.153125	0.065625
6	1.1	1.3	1.5	28	0.109375	0.142188	0.142188	0.164063
7	2.1	3	4	8	0.031250	0.093750	0.093750	0.125000
8	5.6	13	29	1	0.003906	0.050781	0.050781	0.113281
Total				256	1.000000	0.989063	0.989063	0.990625

The final table shows the return for all number of rows and risk levels.

BGAMING Summary

Rows	Low Risk Return	Medium Risk Return	High Risk Return
8	98.91%	98.91%	99.06%
9	99.14%	99.14%	99.06%
10	98.91%	98.91%	99.06%
11	99.02%	99.02%	99.16%
12	98.99%	98.99%	99.12%
13	98.99%	98.99%	99.09%
14	98.99%	98.99%	98.98%
15	99.00%	99.00%	99.03%
16	98.99%	98.99%	98.98%

BetFury

BetFury has their own proprietary Plinko game. It has 16 rows and the pays are similar to the 16-row BGaming game. They are exactly the same between the BGaming "high risk" and the BetFury Red pays.

The following table shows what all three betting options pay, probability of winning, and return to the contribution.

BetFury Plinko

Position	Blue Pays	Green Pays	Red Pays	Combinations	Probability	Low Risk Return	Medium Risk Return	High Risk Return
0	16	110	1000	1	0.000015	0.001678	0.001678	0.015259
1	5	41	130	16	0.000244	0.010010	0.010010	0.031738
2	2	10	26	120	0.001831	0.018311	0.018311	0.047607
3	1.3	5	9	560	0.008545	0.042725	0.042725	0.076904
4	1.2	2.8	4	1,820	0.027771	0.077759	0.077759	0.111084
5	0.2	1.5	2	4,368	0.066650	0.099976	0.099976	0.133301
6	1.1	1	0.2	8,008	0.122192	0.122192	0.122192	0.024438
7	1.1	0.5	0.2	11,440	0.174561	0.087280	0.087280	0.034912
8	1	0.3	0.2	12,870	0.196381	0.058914	0.058914	0.039276
9	1.1	0.5	0.2	11,440	0.174561	0.087280	0.087280	0.034912
10	1.1	1	0.2	8,008	0.122192	0.122192	0.122192	0.024438
11	0.2	1.5	2	4,368	0.066650	0.099976	0.099976	0.133301
12	1.2	2.8	4	1,820	0.027771	0.077759	0.077759	0.111084
13	1.3	5	9	560	0.008545	0.042725	0.042725	0.076904
14	2	10	26	120	0.001831	0.018311	0.018311	0.047607
15	5	41	130	16	0.000244	0.010010	0.010010	0.031738
16	16	110	1000	1	0.000015	0.001678	0.001678	0.015259
Total				65,536	1.000000	0.978775	0.978775	0.989764

The bottom row shows the following overall return by bet:

Blue: 97.88%
Green: 97.88%
Red: 98.98%

Written by: Michael Shackleford

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Plinko

Introduction

Provably Fair Casinos

CryptoGames Rules

Plinko Pay Table

Analysis

Green Pay Table Analysis

Red Pay Table Analysis

Blue Pay Table Analysis

Yellow Pay Table Analysis

Summary

Summary

Probably Fair

BGAMING Rules

16 Rows

15 Rows

14 Rows

13 Rows

12 Rows

11 Rows

10 Rows

9 Rows

8 Rows

BGAMING Summary

BetFury

BetFury Plinko