Pass the pigs

Version francaise
Elementary probabilities
Probabilities after several turns
Constant point strategy
Links to other sites

Introduction

Pass the pigs is a game edited by Winning Moves, where you throw little plastic pigs instead of dices. This page describes probabilities found and several strategies.

Elementary probabilities

Here are results found when 844 pigs were thrown together, that is 1688 (pigs alone) :

Pink side	0.36
Black dot side	0.27
Trotter	0.07
Razorback	0.24
Snouter	0.02
Leaning Jowler	0.01

If we suppose that results are the same for one or two pigs (to check) we get the following results:

Double Pink Side	0.132741
Pink Side Black dot side	0.203320
Pink Side Trotter	0.055255
Pink Side Razorback	0.177420
Pink Side Snouter	0.019857
Pink Side Leaning Jowler	0.007339
Double Black dot side	0.077857
Black dot side Trotter	0.042317
Black dot side Razorback	0.135878
Black dot side Snouter	0.015208
Black dot side Leaning Jowler	0.005620
Double Trotter	0.005750
Trotter Razorback	0.036926
Trotter Snouter	0.004133
Trotter Leaning Jowler	0.001527
Double Razorback	0.059284
Razorback Snouter	0.013270
Razorback Leaning Jowler	0.004904
Double Snouter	0.000743
Snouter Leaning Jowler	0.000549
Double Leaning Jowler	0.000101

We denote by p₀=0.203 probability to have a null score.

Expected value

Expected value or expectation is the sum of the values of a random variable (like the number of points to get) with a weight proportional to the probability of the occurrence of the value. Given previous results we can compute expectation of points when throwing two pigs: E=4.919 points.

Probability to get to turn N is then (1-p₀)^N. Then the expectation of points while playing continuously is: E/p₀ which is 24.23 points.

Expectations after several couple of pigs thrown

Here are probabilities after several couple of pigs were thrown. Probability to have a null score is not represented.

One couple of pigs thrown

Two couples of pigs thrown

Three couples of pigs thrown

Seven Couples of pigs thrown

Constant point strategy

A natural strategy is to have a fixed number of points and to stop when you reach a given number. For instance "whatever my opponent score is, I shall stop when I reach 20 points". 20 is named the threshold value.

Table below shows results for several threshold. Each table cell represent 20 000 plays between strategies. Cell (17x20) represents the result of the plays between threshold strategy "stop at 17 points" against strategy "stop at 20 points". The number in the cell (here 0.95) means that strategy "stop at 17 points" has won 0.95 times less than strategy "stop at 20 points". 9.3 millions plays were computed to generate the table.

Since the table is symmetric, only the upper part has been represented. Diagonal figures should be 1, but their are not. This gives an idea of the precision of computation.
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
10 1 0.74 0.65 0.62 0.61 0.6 0.58 0.52 0.52 0.51 0.53 0.52 0.5 0.5 0.5 0.5 0.57 0.59 0.59 0.59 0.59 0.57 0.57 0.6 0.62 0.62 0.72 0.82 0.88 0.93 0.96 10

11 1 0.88 0.83 0.83 0.8 0.74 0.7 0.69 0.68 0.7 0.67 0.67 0.64 0.64 0.63 0.71 0.74 0.74 0.73 0.74 0.72 0.69 0.75 0.79 0.76 0.87 0.99 1.1 1.1 1.1 11

12 1 0.96 0.95 0.92 0.86 0.81 0.78 0.76 0.8 0.79 0.74 0.74 0.72 0.72 0.8 0.82 0.83 0.82 0.84 0.82 0.78 0.81 0.85 0.86 0.99 1.1 1.2 1.2 1.3 12

13 0.98 0.97 0.97 0.91 0.86 0.82 0.83 0.84 0.78 0.78 0.75 0.76 0.78 0.85 0.86 0.85 0.86 0.86 0.84 0.82 0.83 0.87 0.88 1 1.1 1.2 1.3 1.3 13

14 1 0.99 0.92 0.87 0.83 0.83 0.83 0.83 0.8 0.79 0.77 0.78 0.85 0.88 0.87 0.86 0.88 0.82 0.8 0.84 0.89 0.91 1 1.2 1.2 1.3 1.3 14

15 1 0.92 0.84 0.83 0.85 0.83 0.82 0.8 0.77 0.78 0.79 0.87 0.9 0.87 0.89 0.88 0.83 0.82 0.85 0.89 0.92 1 1.2 1.2 1.3 1.3 15

16 1 0.95 0.91 0.93 0.92 0.91 0.84 0.85 0.83 0.86 0.93 0.94 0.93 0.96 0.94 0.89 0.88 0.94 0.97 0.98 1.1 1.2 1.3 1.3 1.4 16

17 0.97 0.97 0.96 0.95 0.93 0.88 0.9 0.89 0.88 0.96 0.99 0.99 0.99 0.98 0.96 0.91 0.96 1 1 1.2 1.3 1.4 1.4 1.4 17

18 0.99 0.98 0.99 0.97 0.94 0.91 0.93 0.92 0.98 1 1 1 1 0.98 0.94 0.96 1 1 1.2 1.3 1.4 1.4 1.5 18

19 0.99 0.97 0.97 0.97 0.91 0.9 0.93 0.99 1 1 1 1 0.97 0.95 0.99 1 1 1.2 1.3 1.4 1.4 1.5 19

20 0.98 1 0.95 0.93 0.9 0.92 1 1 1 1 1 0.95 0.96 1 1 1.1 1.2 1.3 1.4 1.5 1.4 20

21 1 0.95 0.94 0.93 0.93 1 1 1 1 1 0.97 0.97 1 1 1.1 1.2 1.3 1.4 1.5 1.5 21

22 1 1 0.97 0.98 1.1 1.1 1.1 1.1 1.1 1 0.98 1.1 1.1 1.1 1.2 1.4 1.5 1.5 1.6 22

23 1 0.97 1 1.1 1.1 1.1 1.1 1.1 1 1 1 1.1 1.1 1.2 1.4 1.5 1.5 1.5 23

24 1 1 1.1 1.1 1.1 1.1 1.1 1 1 1 1.1 1.1 1.3 1.4 1.5 1.5 1.5 24

25 1 1.1 1.1 1.1 1.1 1.1 1 1 1.1 1.1 1.1 1.2 1.4 1.4 1.5 1.5 25

26 0.98 1 1 0.98 1 0.94 0.96 0.98 1 1 1.2 1.3 1.3 1.4 1.5 26

27 1 1 0.97 1 0.96 0.94 0.95 1 1 1.1 1.3 1.4 1.4 1.4 27

28 1 0.98 1 0.97 0.92 0.96 1 1 1.1 1.3 1.3 1.4 1.4 28

29 1 1 0.94 0.94 0.99 1 1 1.1 1.3 1.3 1.4 1.5 29

30 1 0.95 0.98 1 1 1 1.1 1.2 1.4 1.4 1.4 30

31 0.98 0.98 1 1.1 1.1 1.2 1.3 1.4 1.4 1.5 31

32 1 1 1.1 1.1 1.2 1.3 1.4 1.5 1.5 32

33 0.99 1 1 1.2 1.3 1.4 1.4 1.4 33

34 1 0.99 1.1 1.2 1.3 1.4 1.4 34

35 0.99 1.1 1.2 1.3 1.3 1.4 35

36 1 1.1 1.2 1.2 1.2 36

37 1 1 1.1 1.1 37

38 0.99 1 1 38

39 0.99 1 39

40 1 40

     <= 0.8 Very bad for vertical strategy

     0.8 <...< 0.96 Bad for vertical strategy

     0.96 <...< 1.05 Neutral

     1.05 <...< 1.1 Good for vertical strategy

     >=1.1 Very good for vertical strategy

Node: displayed figures are rounded. This is why some 1.1 figure appear on dark green and some other on light green.

Stop and risk strategy

Stopping at a given threshold like in previous strategy sometimes lakes taste of risk. This strategy is also a threshold strategy but is different: keep throwing pigs as long as the total score is not above the opponent and the threshold has not been reached.

This technique (Stop_and_risk) has been compared to previous strategy (Stop_at) for thresholds from 10 to 40. ratios have been given for 10 000 plays in each cell. Stop_at are horizontal, Stop_and_risk are vertical.

	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40
10	1.6	1.2	1.1	1	1	0.99	0.9	0.86	0.83	0.82	0.84	0.83	0.79	0.77	0.75	0.76	0.86	0.86	0.86	0.85	0.84	0.77	0.76	0.77	0.8	0.83	0.91	1.1	1.2	1.2	1.3	10
11	1.8	1.4	1.2	1.1	1.2	1.1	1.1	0.99	0.93	0.97	0.95	0.94	0.87	0.84	0.86	0.85	0.96	0.96	0.95	0.95	0.94	0.86	0.87	0.86	0.92	0.93	1.1	1.2	1.3	1.4	1.3	11
12	2	1.5	1.3	1.2	1.2	1.2	1.1	1	1	1	1.1	0.99	0.91	0.92	0.91	0.92	1	1	0.98	1	0.98	0.94	0.87	0.94	0.99	0.98	1.1	1.2	1.3	1.4	1.4	12
13	2.1	1.6	1.4	1.3	1.3	1.2	1.1	1.1	1	1	1	0.99	0.95	0.94	0.95	0.94	1	1	1	1	1	0.95	0.92	0.94	1	1	1.1	1.3	1.4	1.5	1.5	13
14	2.1	1.5	1.4	1.3	1.3	1.3	1.1	1.1	1.1	1	1.1	1	0.98	0.97	0.96	0.93	1	1	1	1	1	0.96	0.95	0.95	1	1	1.1	1.3	1.4	1.4	1.5	14
15	2	1.6	1.3	1.3	1.2	1.3	1.1	1.1	1.1	1	1.1	1	0.98	0.93	0.93	0.97	1	1	1	1	1	0.92	0.91	0.99	1	1.1	1.1	1.3	1.4	1.4	1.5	15
16	2.1	1.6	1.4	1.3	1.3	1.3	1.2	1.2	1.1	1.1	1.1	1.1	1	0.97	1	1	1.1	1.1	1.1	1	1.1	1	0.97	1	1	1.1	1.2	1.3	1.4	1.5	1.5	16
17	2.2	1.6	1.4	1.4	1.3	1.4	1.2	1.1	1.1	1.1	1.1	1.1	1	0.97	1	1	1.1	1.1	1.1	1.1	1.1	1	0.99	1	1.1	1.1	1.2	1.4	1.5	1.5	1.5	17
18	2.2	1.6	1.4	1.3	1.4	1.4	1.2	1.1	1.1	1.1	1.1	1.1	1	1	1	1	1.1	1.1	1.1	1.1	1	1	1	1.1	1.1	1.1	1.2	1.4	1.5	1.5	1.5	18
19	2.2	1.7	1.4	1.4	1.4	1.4	1.2	1.2	1.1	1.1	1.1	1.1	1	1	1	1	1.1	1.1	1.1	1.1	1.1	1	1	1	1.1	1.1	1.2	1.4	1.5	1.5	1.5	19
20	2.2	1.6	1.5	1.4	1.3	1.4	1.2	1.1	1.1	1.1	1.1	1.1	1.1	1	1	1	1.1	1.1	1.1	1.1	1.1	1	1	1	1.1	1.1	1.2	1.3	1.4	1.5	1.5	20
21	2.1	1.7	1.4	1.4	1.3	1.3	1.2	1.1	1.1	1.1	1.1	1.1	1	1	1	1	1.1	1.1	1.1	1.1	1.1	1	0.99	1	1.1	1.1	1.2	1.3	1.5	1.5	1.5	21
22	2.1	1.6	1.4	1.4	1.3	1.3	1.2	1.2	1.1	1.1	1.1	1.1	1	1	1	1	1.1	1.1	1.1	1.1	1.1	1	1	1.1	1.1	1.1	1.2	1.3	1.4	1.5	1.5	22
23	2.1	1.6	1.4	1.4	1.3	1.4	1.2	1.1	1.1	1.1	1.1	1.1	1	1	1	0.99	1.1	1.1	1.1	1.1	1.1	1.1	1	1	1.1	1.1	1.2	1.4	1.4	1.4	1.5	23
24	2.1	1.6	1.4	1.4	1.3	1.4	1.2	1.1	1.1	1.1	1.1	1.1	1	1	1	1	1.1	1.1	1.1	1.1	1.1	1.1	1	1	1.1	1.1	1.2	1.3	1.4	1.4	1.5	24
25	2	1.6	1.4	1.3	1.3	1.4	1.3	1.2	1.1	1.1	1.1	1.1	1	1	1	0.98	1.1	1.1	1.1	1.1	1.1	1	1	1.1	1.1	1.1	1.2	1.3	1.5	1.4	1.4	25
26	1.8	1.5	1.3	1.2	1.2	1.2	1.2	1.1	1	1.1	1.1	1	0.99	0.95	0.95	0.94	1	1.1	1.1	1	1	1	1	1	1.1	1.1	1.1	1.2	1.3	1.4	1.4	26
27	1.8	1.4	1.2	1.2	1.2	1.2	1.1	1.1	1	1	1	1	0.95	0.95	0.93	0.93	0.99	1	1	1	1	0.97	0.95	0.98	1	1	1.1	1.2	1.3	1.3	1.3	27
28	1.8	1.3	1.3	1.2	1.2	1.2	1.1	1	1	1	1	0.99	0.95	0.93	0.93	0.91	0.98	0.99	1	1	1	0.98	0.95	0.99	1	1	1.1	1.2	1.3	1.3	1.4	28
29	1.7	1.4	1.3	1.2	1.2	1.2	1.1	1.1	1	1	1	1	0.96	0.92	0.93	0.92	1	1	1	1	1	0.98	0.96	1	1	1	1.1	1.2	1.3	1.3	1.3	29
30	1.7	1.4	1.3	1.2	1.2	1.2	1.1	1	1	1	1	1	0.93	0.93	0.93	0.94	1	1	1	1	1	0.96	0.97	0.98	1	1	1.1	1.2	1.3	1.3	1.3	30
31	1.8	1.4	1.2	1.2	1.2	1.2	1.1	1	1	1	1	0.99	0.98	0.96	0.97	0.96	1	1	1	1	1	0.97	0.95	0.98	1	1	1.1	1.2	1.3	1.3	1.3	31
32	1.7	1.3	1.2	1.2	1.2	1.2	1.1	1	1	1	1	1	0.98	0.92	0.95	0.95	1	1	1	1	1	0.96	0.94	0.96	1	1	1.1	1.2	1.2	1.3	1.3	32
33	1.6	1.3	1.2	1.1	1.2	1.1	1.1	1	0.99	0.98	0.96	0.95	0.92	0.9	0.88	0.91	0.97	1	1	0.98	0.99	0.93	0.92	0.92	0.95	0.99	1.1	1.1	1.2	1.3	1.2	33
34	1.6	1.3	1.2	1.1	1.1	1.1	1	0.97	0.95	0.93	0.94	0.94	0.91	0.89	0.89	0.86	0.95	0.98	0.94	0.95	0.95	0.9	0.89	0.91	0.97	0.95	1	1.1	1.2	1.2	1.3	34
35	1.6	1.2	1.1	1.1	1.1	1.1	1	0.95	0.96	0.93	0.93	0.92	0.9	0.86	0.87	0.87	0.93	0.96	0.95	0.96	0.94	0.89	0.9	0.88	0.93	0.95	1	1.1	1.1	1.2	1.2	35
36	1.4	1.1	1	1	0.97	1	0.94	0.86	0.86	0.84	0.85	0.84	0.81	0.81	0.78	0.81	0.87	0.88	0.87	0.87	0.87	0.84	0.84	0.84	0.86	0.86	0.95	1	1.1	1.1	1.1	36
37	1.3	1	0.94	0.9	0.9	0.88	0.83	0.8	0.79	0.8	0.79	0.78	0.76	0.73	0.74	0.74	0.78	0.81	0.8	0.82	0.81	0.77	0.75	0.76	0.77	0.8	0.86	0.96	1	1	1.1	37
38	1.2	0.98	0.9	0.85	0.84	0.83	0.78	0.77	0.74	0.75	0.72	0.73	0.71	0.69	0.69	0.69	0.76	0.78	0.75	0.76	0.75	0.71	0.71	0.73	0.73	0.77	0.83	0.9	1	0.98	1	38
39	1.1	0.93	0.84	0.8	0.82	0.82	0.77	0.75	0.72	0.72	0.72	0.71	0.68	0.68	0.69	0.68	0.72	0.74	0.75	0.74	0.74	0.72	0.69	0.71	0.72	0.72	0.83	0.9	0.95	0.97	0.97	39
40	1.1	0.94	0.85	0.82	0.82	0.8	0.74	0.73	0.72	0.69	0.71	0.7	0.68	0.67	0.67	0.69	0.72	0.74	0.74	0.74	0.71	0.71	0.7	0.7	0.72	0.73	0.78	0.88	0.94	0.97	1	40

Links to related sites

Optimal strategy for the pig game, I also recommend the following article by T. Neller and G. M. Presser Optimal Play of the Dice Game Pig, The UMAP Journal 25(1) (2004), pp. 25-47. Their home page.
Another probability.

If you have any questions: Fabrice Derepas

derepas.com
Last modified: Thu Nov 4 09:44:01 2004

	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40
10	1	0.74	0.65	0.62	0.61	0.6	0.58	0.52	0.52	0.51	0.53	0.52	0.5	0.5	0.5	0.5	0.57	0.59	0.59	0.59	0.59	0.57	0.57	0.6	0.62	0.62	0.72	0.82	0.88	0.93	0.96	10
11		1	0.88	0.83	0.83	0.8	0.74	0.7	0.69	0.68	0.7	0.67	0.67	0.64	0.64	0.63	0.71	0.74	0.74	0.73	0.74	0.72	0.69	0.75	0.79	0.76	0.87	0.99	1.1	1.1	1.1	11
12			1	0.96	0.95	0.92	0.86	0.81	0.78	0.76	0.8	0.79	0.74	0.74	0.72	0.72	0.8	0.82	0.83	0.82	0.84	0.82	0.78	0.81	0.85	0.86	0.99	1.1	1.2	1.2	1.3	12
13				0.98	0.97	0.97	0.91	0.86	0.82	0.83	0.84	0.78	0.78	0.75	0.76	0.78	0.85	0.86	0.85	0.86	0.86	0.84	0.82	0.83	0.87	0.88	1	1.1	1.2	1.3	1.3	13
14					1	0.99	0.92	0.87	0.83	0.83	0.83	0.83	0.8	0.79	0.77	0.78	0.85	0.88	0.87	0.86	0.88	0.82	0.8	0.84	0.89	0.91	1	1.2	1.2	1.3	1.3	14
15						1	0.92	0.84	0.83	0.85	0.83	0.82	0.8	0.77	0.78	0.79	0.87	0.9	0.87	0.89	0.88	0.83	0.82	0.85	0.89	0.92	1	1.2	1.2	1.3	1.3	15
16							1	0.95	0.91	0.93	0.92	0.91	0.84	0.85	0.83	0.86	0.93	0.94	0.93	0.96	0.94	0.89	0.88	0.94	0.97	0.98	1.1	1.2	1.3	1.3	1.4	16
17								0.97	0.97	0.96	0.95	0.93	0.88	0.9	0.89	0.88	0.96	0.99	0.99	0.99	0.98	0.96	0.91	0.96	1	1	1.2	1.3	1.4	1.4	1.4	17
18									0.99	0.98	0.99	0.97	0.94	0.91	0.93	0.92	0.98	1	1	1	1	0.98	0.94	0.96	1	1	1.2	1.3	1.4	1.4	1.5	18
19										0.99	0.97	0.97	0.97	0.91	0.9	0.93	0.99	1	1	1	1	0.97	0.95	0.99	1	1	1.2	1.3	1.4	1.4	1.5	19
20											0.98	1	0.95	0.93	0.9	0.92	1	1	1	1	1	0.95	0.96	1	1	1.1	1.2	1.3	1.4	1.5	1.4	20
21												1	0.95	0.94	0.93	0.93	1	1	1	1	1	0.97	0.97	1	1	1.1	1.2	1.3	1.4	1.5	1.5	21
22													1	1	0.97	0.98	1.1	1.1	1.1	1.1	1.1	1	0.98	1.1	1.1	1.1	1.2	1.4	1.5	1.5	1.6	22
23														1	0.97	1	1.1	1.1	1.1	1.1	1.1	1	1	1	1.1	1.1	1.2	1.4	1.5	1.5	1.5	23
24															1	1	1.1	1.1	1.1	1.1	1.1	1	1	1	1.1	1.1	1.3	1.4	1.5	1.5	1.5	24
25																1	1.1	1.1	1.1	1.1	1.1	1	1	1.1	1.1	1.1	1.2	1.4	1.4	1.5	1.5	25
26																	0.98	1	1	0.98	1	0.94	0.96	0.98	1	1	1.2	1.3	1.3	1.4	1.5	26
27																		1	1	0.97	1	0.96	0.94	0.95	1	1	1.1	1.3	1.4	1.4	1.4	27
28																			1	0.98	1	0.97	0.92	0.96	1	1	1.1	1.3	1.3	1.4	1.4	28
29																				1	1	0.94	0.94	0.99	1	1	1.1	1.3	1.3	1.4	1.5	29
30																					1	0.95	0.98	1	1	1	1.1	1.2	1.4	1.4	1.4	30
31																						0.98	0.98	1	1.1	1.1	1.2	1.3	1.4	1.4	1.5	31
32																							1	1	1.1	1.1	1.2	1.3	1.4	1.5	1.5	32
33																								0.99	1	1	1.2	1.3	1.4	1.4	1.4	33
34																									1	0.99	1.1	1.2	1.3	1.4	1.4	34
35																										0.99	1.1	1.2	1.3	1.3	1.4	35
36																											1	1.1	1.2	1.2	1.2	36
37																												1	1	1.1	1.1	37
38																													0.99	1	1	38
39																														0.99	1	39
40																															1	40