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.

10111213141516171819202122232425262728293031323334353637383940
1010.740.650.620.610.60.580.520.520.510.530.520.50.50.50.50.570.590.590.590.590.570.570.60.620.620.720.820.880.930.9610
1110.880.830.830.80.740.70.690.680.70.670.670.640.640.630.710.740.740.730.740.720.690.750.790.760.870.991.11.11.111
1210.960.950.920.860.810.780.760.80.790.740.740.720.720.80.820.830.820.840.820.780.810.850.860.991.11.21.21.312
130.980.970.970.910.860.820.830.840.780.780.750.760.780.850.860.850.860.860.840.820.830.870.8811.11.21.31.313
1410.990.920.870.830.830.830.830.80.790.770.780.850.880.870.860.880.820.80.840.890.9111.21.21.31.314
1510.920.840.830.850.830.820.80.770.780.790.870.90.870.890.880.830.820.850.890.9211.21.21.31.315
1610.950.910.930.920.910.840.850.830.860.930.940.930.960.940.890.880.940.970.981.11.21.31.31.416
170.970.970.960.950.930.880.90.890.880.960.990.990.990.980.960.910.96111.21.31.41.41.417
180.990.980.990.970.940.910.930.920.9811110.980.940.96111.21.31.41.41.518
190.990.970.970.970.910.90.930.9911110.970.950.99111.21.31.41.41.519
200.9810.950.930.90.92111110.950.96111.11.21.31.41.51.420
2110.950.940.930.93111110.970.97111.11.21.31.41.51.521
22110.970.981.11.11.11.11.110.981.11.11.11.21.41.51.51.622
2310.9711.11.11.11.11.11111.11.11.21.41.51.51.523
24111.11.11.11.11.11111.11.11.31.41.51.51.524
2511.11.11.11.11.1111.11.11.11.21.41.41.51.525
260.98110.9810.940.960.98111.21.31.31.41.526
27110.9710.960.940.95111.11.31.41.41.427
2810.9810.970.920.96111.11.31.31.41.428
29110.940.940.99111.11.31.31.41.529
3010.950.981111.11.21.41.41.430
310.980.9811.11.11.21.31.41.41.531
32111.11.11.21.31.41.51.532
330.99111.21.31.41.41.433
3410.991.11.21.31.41.434
350.991.11.21.31.31.435
3611.11.21.21.236
37111.11.137
380.991138
390.99139
40140
     <= 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
     >=1.1Very 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.

10111213141516171819202122232425262728293031323334353637383940
101.61.21.1110.990.90.860.830.820.840.830.790.770.750.760.860.860.860.850.840.770.760.770.80.830.911.11.21.21.310
111.81.41.21.11.21.11.10.990.930.970.950.940.870.840.860.850.960.960.950.950.940.860.870.860.920.931.11.21.31.41.311
1221.51.31.21.21.21.11111.10.990.910.920.910.92110.9810.980.940.870.940.990.981.11.21.31.41.412
132.11.61.41.31.31.21.11.11110.990.950.940.950.94111110.950.920.94111.11.31.41.51.513
142.11.51.41.31.31.31.11.11.111.110.980.970.960.93111110.960.950.95111.11.31.41.41.514
1521.61.31.31.21.31.11.11.111.110.980.930.930.97111110.920.910.9911.11.11.31.41.41.515
162.11.61.41.31.31.31.21.21.11.11.11.110.97111.11.11.111.110.97111.11.21.31.41.51.516
172.21.61.41.41.31.41.21.11.11.11.11.110.97111.11.11.11.11.110.9911.11.11.21.41.51.51.517
182.21.61.41.31.41.41.21.11.11.11.11.111111.11.11.11.11111.11.11.11.21.41.51.51.518
192.21.71.41.41.41.41.21.21.11.11.11.111111.11.11.11.11.11111.11.11.21.41.51.51.519
202.21.61.51.41.31.41.21.11.11.11.11.11.11111.11.11.11.11.11111.11.11.21.31.41.51.520
212.11.71.41.41.31.31.21.11.11.11.11.111111.11.11.11.11.110.9911.11.11.21.31.51.51.521
222.11.61.41.41.31.31.21.21.11.11.11.111111.11.11.11.11.1111.11.11.11.21.31.41.51.522
232.11.61.41.41.31.41.21.11.11.11.11.11110.991.11.11.11.11.11.1111.11.11.21.41.41.41.523
242.11.61.41.41.31.41.21.11.11.11.11.111111.11.11.11.11.11.1111.11.11.21.31.41.41.524
2521.61.41.31.31.41.31.21.11.11.11.11110.981.11.11.11.11.1111.11.11.11.21.31.51.41.425
261.81.51.31.21.21.21.21.111.11.110.990.950.950.9411.11.1111111.11.11.11.21.31.41.426
271.81.41.21.21.21.21.11.111110.950.950.930.930.9911110.970.950.98111.11.21.31.31.327
281.81.31.31.21.21.21.111110.990.950.930.930.910.980.991110.980.950.99111.11.21.31.31.428
291.71.41.31.21.21.21.11.111110.960.920.930.92111110.980.961111.11.21.31.31.329
301.71.41.31.21.21.21.1111110.930.930.930.94111110.960.970.98111.11.21.31.31.330
311.81.41.21.21.21.21.111110.990.980.960.970.96111110.970.950.98111.11.21.31.31.331
321.71.31.21.21.21.21.1111110.980.920.950.95111110.960.940.96111.11.21.21.31.332
331.61.31.21.11.21.11.110.990.980.960.950.920.90.880.910.97110.980.990.930.920.920.950.991.11.11.21.31.233
341.61.31.21.11.11.110.970.950.930.940.940.910.890.890.860.950.980.940.950.950.90.890.910.970.9511.11.21.21.334
351.61.21.11.11.11.110.950.960.930.930.920.90.860.870.870.930.960.950.960.940.890.90.880.930.9511.11.11.21.235
361.41.1110.9710.940.860.860.840.850.840.810.810.780.810.870.880.870.870.870.840.840.840.860.860.9511.11.11.136
371.310.940.90.90.880.830.80.790.80.790.780.760.730.740.740.780.810.80.820.810.770.750.760.770.80.860.96111.137
381.20.980.90.850.840.830.780.770.740.750.720.730.710.690.690.690.760.780.750.760.750.710.710.730.730.770.830.910.98138
391.10.930.840.80.820.820.770.750.720.720.720.710.680.680.690.680.720.740.750.740.740.720.690.710.720.720.830.90.950.970.9739
401.10.940.850.820.820.80.740.730.720.690.710.70.680.670.670.690.720.740.740.740.710.710.70.70.720.730.780.880.940.97140

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