Statistics

Parrondo's Paradox Calculator

Enter the win probabilities for Game A and the two states of Game B, then pick a playing strategy. The calculator uses Markov chain analysis to find the long-run win probability and expected capital drift for each strategy, proving that two individually losing games can produce a winning combination.

By Dr. Hannah Brandt, PhD · Updated June 25, 2026

Your details

Playing strategy

Game A win probability (p)

Game B win prob when capital mod 3 = 0 (p1)

Game B win prob otherwise (p2)

Starting capital

Rounds to project

rounds

Selected strategy win probabilityWinning strategy

0.508%

Long-run probability of winning each round under the chosen strategy

Expected drift per round0.0157$

Game A win probability0.495%

Game B win probability (Markov)0.496%

Game B drift per round-0.0087$

Game A drift per round-0.01$

Steady-state: P(capital mod 3 = 0)0.35%

Steady-state: P(capital mod 3 = 1)0.25%

Steady-state: P(capital mod 3 = 2)0.4%

Game A drift ($/round)-0.01

Game B drift ($/round)-0.0087

Strategy drift ($/round)0.0157

Round

Expected capital ($)
Round	Game A only	Game B only	Random 50/50 mix
0	100	100	100
1	99.99	99.99	100.02
2	99.98	99.98	100.03
3	99.97	99.97	100.05
4	99.96	99.97	100.06
5	99.95	99.96	100.08
6	99.94	99.95	100.09
7	99.93	99.94	100.11
8	99.92	99.93	100.13
9	99.91	99.92	100.14
10	99.9	99.91	100.16
11	99.89	99.9	100.17
12	99.88	99.9	100.19
13	99.87	99.89	100.2
14	99.86	99.88	100.22
15	99.85	99.87	100.24
16	99.84	99.86	100.25
17	99.83	99.85	100.27
18	99.82	99.84	100.28
19	99.81	99.83	100.3
20	99.8	99.83	100.31
21	99.79	99.82	100.33
22	99.78	99.81	100.35
23	99.77	99.8	100.36
24	99.76	99.79	100.38
25	99.75	99.78	100.39
26	99.74	99.77	100.41
27	99.73	99.77	100.42
28	99.72	99.76	100.44
29	99.71	99.75	100.46
30	99.7	99.74	100.47
31	99.69	99.73	100.49
32	99.68	99.72	100.5
33	99.67	99.71	100.52
34	99.66	99.7	100.53
35	99.65	99.7	100.55
36	99.64	99.69	100.57
37	99.63	99.68	100.58
38	99.62	99.67	100.6
39	99.61	99.66	100.61
40	99.6	99.65	100.63
41	99.59	99.64	100.64
42	99.58	99.63	100.66
43	99.57	99.63	100.68
44	99.56	99.62	100.69
45	99.55	99.61	100.71
46	99.54	99.6	100.72
47	99.53	99.59	100.74
48	99.52	99.58	100.75
49	99.51	99.57	100.77
50	99.5	99.57	100.79
51	99.49	99.56	100.8
52	99.48	99.55	100.82
53	99.47	99.54	100.83
54	99.46	99.53	100.85
55	99.45	99.52	100.86
56	99.44	99.51	100.88
57	99.43	99.5	100.9
58	99.42	99.5	100.91
59	99.41	99.49	100.93
60	99.4	99.48	100.94
61	99.39	99.47	100.96
62	99.38	99.46	100.97
63	99.37	99.45	100.99
64	99.36	99.44	101.01
65	99.35	99.43	101.02
66	99.34	99.43	101.04
67	99.33	99.42	101.05
68	99.32	99.41	101.07
69	99.31	99.4	101.08
70	99.3	99.39	101.1
71	99.29	99.38	101.12
72	99.28	99.37	101.13
73	99.27	99.37	101.15
74	99.26	99.36	101.16
75	99.25	99.35	101.18
76	99.24	99.34	101.19
77	99.23	99.33	101.21
78	99.22	99.32	101.22
79	99.21	99.31	101.24
80	99.2	99.3	101.26
81	99.19	99.3	101.27
82	99.18	99.29	101.29
83	99.17	99.28	101.3
84	99.16	99.27	101.32
85	99.15	99.26	101.33
86	99.14	99.25	101.35
87	99.13	99.24	101.37
88	99.12	99.23	101.38
89	99.11	99.23	101.4
90	99.1	99.22	101.41
91	99.09	99.21	101.43
92	99.08	99.2	101.44
93	99.07	99.19	101.46
94	99.06	99.18	101.48
95	99.05	99.17	101.49
96	99.04	99.17	101.51
97	99.03	99.16	101.52
98	99.02	99.15	101.54
99	99.01	99.14	101.55
100	99	99.13	101.57
101	98.99	99.12	101.59
102	98.98	99.11	101.6
103	98.97	99.1	101.62
104	98.96	99.1	101.63
105	98.95	99.09	101.65
106	98.94	99.08	101.66
107	98.93	99.07	101.68
108	98.92	99.06	101.7
109	98.91	99.05	101.71
110	98.9	99.04	101.73
111	98.89	99.03	101.74
112	98.88	99.03	101.76
113	98.87	99.02	101.77
114	98.86	99.01	101.79
115	98.85	99	101.81
116	98.84	98.99	101.82
117	98.83	98.98	101.84
118	98.82	98.97	101.85
119	98.81	98.97	101.87
120	98.8	98.96	101.88
121	98.79	98.95	101.9
122	98.78	98.94	101.92
123	98.77	98.93	101.93
124	98.76	98.92	101.95
125	98.75	98.91	101.96
126	98.74	98.9	101.98
127	98.73	98.9	101.99
128	98.72	98.89	102.01
129	98.71	98.88	102.03
130	98.7	98.87	102.04
131	98.69	98.86	102.06
132	98.68	98.85	102.07
133	98.67	98.84	102.09
134	98.66	98.83	102.1
135	98.65	98.83	102.12
136	98.64	98.82	102.14
137	98.63	98.81	102.15
138	98.62	98.8	102.17
139	98.61	98.79	102.18
140	98.6	98.78	102.2
141	98.59	98.77	102.21
142	98.58	98.77	102.23
143	98.57	98.76	102.25
144	98.56	98.75	102.26
145	98.55	98.74	102.28
146	98.54	98.73	102.29
147	98.53	98.72	102.31
148	98.52	98.71	102.32
149	98.51	98.7	102.34
150	98.5	98.7	102.36
151	98.49	98.69	102.37
152	98.48	98.68	102.39
153	98.47	98.67	102.4
154	98.46	98.66	102.42
155	98.45	98.65	102.43
156	98.44	98.64	102.45
157	98.43	98.63	102.47
158	98.42	98.63	102.48
159	98.41	98.62	102.5
160	98.4	98.61	102.51
161	98.39	98.6	102.53
162	98.38	98.59	102.54
163	98.37	98.58	102.56
164	98.36	98.57	102.58
165	98.35	98.57	102.59
166	98.34	98.56	102.61
167	98.33	98.55	102.62
168	98.32	98.54	102.64
169	98.31	98.53	102.65
170	98.3	98.52	102.67
171	98.29	98.51	102.69
172	98.28	98.5	102.7
173	98.27	98.5	102.72
174	98.26	98.49	102.73
175	98.25	98.48	102.75
176	98.24	98.47	102.76
177	98.23	98.46	102.78
178	98.22	98.45	102.8
179	98.21	98.44	102.81
180	98.2	98.43	102.83
181	98.19	98.43	102.84
182	98.18	98.42	102.86
183	98.17	98.41	102.87
184	98.16	98.4	102.89
185	98.15	98.39	102.91
186	98.14	98.38	102.92
187	98.13	98.37	102.94
188	98.12	98.37	102.95
189	98.11	98.36	102.97
190	98.1	98.35	102.98
191	98.09	98.34	103
192	98.08	98.33	103.02
193	98.07	98.32	103.03
194	98.06	98.31	103.05
195	98.05	98.3	103.06
196	98.04	98.3	103.08
197	98.03	98.29	103.09
198	98.02	98.28	103.11
199	98.01	98.27	103.13
200	98	98.26	103.14

Game A only
Game B only
Random 50/50 mix

The paradox works: randomly mixing A and B 50/50 wins at 50.785% despite both games losing alone.

Game A alone has a 49.500% win rate - below 50%, so it loses in the long run (drift: -0.0100 $/round).
Game B alone has a 49.565% win rate via Markov analysis - also below 50% (drift: -0.0087 $/round).
The mixed strategy (randomly mixing A and B 50/50) achieves 50.785% - above 50%! After 200 rounds, expected capital gain: $3.14.
This is Parrondo's Paradox in action: mixing two losing games creates a winning strategy by shifting the system away from the capital-mod-3 trap state.

Next stepTry adjusting p1 lower and p2 higher while keeping both games individually losing to see the paradox emerge more strongly. The effect is maximized when epsilon (the bias away from 0.5) is small.

Game A expected drift per round2 × 0.495 - 1
-0.0100 $/round
Game B steady-state win probability (Markov chain)pi0 × p1 + pi1 × p2 + pi2 × p2
0.49565
Game B steady-state probabilities[pi0, pi1, pi2] from solving pi = pi × T
[0.3451, 0.2541, 0.4008]
Blended win prob per state (50% A, 50% B)state 0: 0.5×0.495+0.5×0.095=0.2950, state 1/2: 0.5×0.495+0.5×0.745=0.6200
w0=0.2950, w1=0.6200, w2=0.6200
Mixed strategy steady-state win probabilitypi0×w0 + pi1×w1 + pi2×w2
0.50785 (winning)
Expected drift under chosen strategy2 × 0.50785 - 1
0.0157 $/round
Projected capital after 200 rounds$100 + 0.0157 × 200
$103.14

What is Parrondo's Paradox?

Parrondo's Paradox, discovered in 1996 by Spanish physicist Juan Manuel Rodriguez Parrondo, is the counterintuitive result that two individually losing gambling games can be combined into a winning strategy. It sounds impossible: if every individual game you play loses money in the long run, how can playing them together make you richer? The answer lies in how the games interact through a shared state, typically your current capital. When you mix the games, you spend less time in the losing states of Game B and more time in the winning states, shifting the overall balance above 50%. The paradox is a rigorously proven mathematical fact, not a trick, and has been applied in biology (genetic drift), economics (portfolio theory), and physics (Brownian ratchets).

How the two games work

Game A is a simple biased coin: you win $1 with probability p (slightly less than 0.5, typically 0.495) and lose $1 otherwise. Playing it alone guarantees a slow drift toward zero. Game B depends on your current capital divided by 3. If your capital is exactly divisible by 3 (the 'trap' state), you face very unfavorable odds (win prob 0.095). If your capital is NOT divisible by 3, you face very favorable odds (win prob 0.745). At first glance Game B seems attractive because you spend most time in the good state. But the Markov chain analysis shows that Game B's steady-state win probability settles around 49.6%, just below 50%, so it too loses in the long run when played alone. The trap state at capital mod 3 = 0 is visited just often enough to drag the average below fair.

Why mixing the games works: the Markov chain explanation

The key insight is that Game A and Game B have opposite effects on the mod-3 state of your capital. Game A moves capital up or down by 1, redistributing the mod-3 state. When you intersperse A and B plays, Game A effectively breaks the cycle that keeps dragging you into Game B's trap state. The Markov chain for a 50/50 random mix has a different steady-state distribution: the fraction of time spent in the trap state (capital mod 3 = 0) drops from about 38.4% to about 34.5%, and the overall win probability rises to about 50.8%. This is enough to flip the sign of the drift from negative to positive. You go from losing roughly 1 cent per round in each game alone to gaining roughly 1.6 cents per round by mixing them. The mathematical tool for computing this is the transition matrix of the combined Markov chain, whose left eigenvector with eigenvalue 1 gives the steady-state probabilities.

How to use this calculator

Enter the win probability for Game A (p), and the two win probabilities for Game B: p1 for the 'trap' state when capital mod 3 equals 0, and p2 for all other states. Choose your playing strategy from the dropdown: A only, B only, strict alternation (ABAB), the AABB pattern, or a random 50/50 mix. The calculator solves the Markov chain analytically and shows the long-run win probability, expected drift per round, and the steady-state distribution of capital mod 3 for your chosen strategy. The chart shows the expected capital trajectory over time for all three scenarios side by side so you can see the paradox visually. To reproduce the classic Parrondo result, use the default values: p=0.495, p1=0.095, p2=0.745.

Classic Parrondo's paradox probability values

Game	Condition	Win probability	Drift per round
Game A	Always	0.495	-0.0100
Game B	Capital mod 3 = 0	0.095	-
Game B	Capital mod 3 != 0	0.745	-
Game B	Long-run (Markov)	~0.49565	-0.00870
A,B mix (random)	Long-run (Markov)	~0.50785	+0.01570

The original game probabilities used by Juan Parrondo. Each game loses alone, but any mixed strategy wins.

Frequently asked questions

Is Parrondo's paradox real or just a trick?

It is mathematically proven. Parrondo's paradox arises from a genuine property of Markov chains: two transition matrices that individually lead to a losing drift can combine into one with a positive drift. It is not an artifact of small samples or incorrect probability calculations. The effect has been demonstrated experimentally with physical ratchets and observed in biology.

Can I use this at a casino?

No. The paradox requires a specific capital-dependent interaction between the two games. Standard casino games are statistically independent of your current capital (the roulette wheel does not know how much money you have). Without that coupling, mixing losing games produces only a weighted average of losing strategies, which still loses.

What is the mod-3 trap state?

Game B's rules depend on whether your capital is divisible by 3. When capital mod 3 equals 0, you face the very bad odds (0.095 win probability). When it equals 1 or 2, you face favorable odds (0.745). The trap is that even with mostly favorable plays, you visit the divisible-by-3 state just often enough to drag the overall expected value below 50%. The Markov chain visits state 0 about 38.4% of the time when playing Game B alone.

Why does alternating between A and B help?

Game A shifts your capital by +1 or -1 with nearly equal probability, which changes the capital mod 3 value randomly. This disrupts the pattern that keeps Game B cycling back to the trap state. Specifically, A plays reduce the fraction of time spent in the mod-3=0 trap state from about 38.4% to about 34.5%, tilting the overall win rate just above 50%.

Do ABAB and AABB give the same result?

In this calculator both are modeled as a 50/50 blend of Game A and Game B steps, which gives identical long-run win probabilities under the blended Markov chain model. In a more detailed simulation, the exact sequence can produce small differences, but both reliably produce the winning paradox for the classic probability values.

What happens if I change the probability values?

The paradox is not guaranteed for all probability combinations. It emerges when: (1) both games individually lose (win prob below 0.5), (2) the capital-dependent structure of Game B creates a strong enough trap, and (3) Game A's uniform random walk disrupts that trap effectively. If the bias is too large or the trap is not strong enough, mixing may not flip the overall result to winning.

What real-world systems exhibit this paradox?

Parrondo's paradox has been identified in genetic drift in biology (where two deleterious mutations can combine to produce a beneficial effect), in models of financial portfolio diversification, in ecology (where two individually declining species stabilize when mixed), and in the physics of Brownian ratchets and molecular motors. It has even been proposed as an explanation for certain evolutionary phenomena where random genetic drift interacts with selection.

Sources

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Written by Dr. Hannah Brandt, PhD Statistician · Munich, Germany

Applied statistician translating rigorous probability theory into clear, accurate tools for researchers and practitioners.

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