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Mathematical Theory Analysis of Sic Bo game

First, the problem statement

Players first select the bet number of chips, then choose to buy big or buy small, determined the three Dice Randomly generated by the system program three random numbers from 1 to 6, if these three numbers are the same, regardless buy big or buy small players are back to deduct the number of chips bet; if different, then these three numbers together 4 to 10 points for the small, 11 to 17 large, if the pressure on the players to get the size of the bet number of chips.

Now thereby raised three questions:

1, buy a big win more or buy small win more?

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2, this wager it is possible to make money?

3. How to play to more money, if there is a play makes money?

Second, simplification and assumptions

Suppose the player has a number of chips as M (M is a natural number)

The number of chips bet no time for N (N> = 1000, N is a natural number)

When buying an hour, let f = -1; when to buy large, set f = 1

Let the three dice as a, b, c (a, b, c is a natural number of 1 to 6)

When a = b = c, that the dealer if the dice roll out a full (three dice the same) size of the take-home, set g = 0;

When a + b + c = 4 ~ 10, that is to open a small, g = -1;

When a + b + c = 11 ~ 17, that is to open a large, g = 1.

h = 1 && f * g = 1 || h = -1 && f * g = 0 | -1

After the 1 Bureau, the number of chips the player is: M + h * N

N Post Office, the number of chips the player is: M + h1 * N1 + h2 * N2 + ... + hn * Nn..

Third, the model and its solution

1, the first inning on a single dice were analyzed

Since the system source code is unknown, can be assumed that each of points 1 to 6 dice appears to be random, then the three dice, the combinations have XXX, XXY, XYZ two kinds, XXX including only one, but also includes XYX XXY , YXX total of three kinds, and there are six kinds of combinations of XYZ, the following table lists start small, take-all, open a large number of species:

Points combinations to open a small take-open big

3111010

4112300

5 600 113, 122

6 114,123,222 910

7 115,124,133,223 1500

8 116,125,134,224,233 2100

9 126,135,144,225,234,333 2410

10 136,145,226,235,244,334 2700

11 146,155,236,245,335,344 0027

12 156,246,255,336,345,444 0124

13 166,256,346,355,445 0021

14 266,356,446,455 0015

15 366,456,555 019

16 466,556 006

17566003

18666010

Total: 1,056,105

A total of three dice combinations 6 * 6 * 6 = 216 kinds

Takes all probability is: 6/216 = 1/36 = 2.78%

Open big probability: 105/216 = 35/72 = 48.61%

Open a small probability: 105/216 = 35/72 = 48.61%

Thus for a separate office, the same small probability open wide open.

Then:

2, the primary player bets:

Beginning usually return this play: each game a certain number of bets for this case is the number of chips bet some N, then after n Bureau, the number of chips the player is:. M + (. H1 + h2 + ... + hn) * N

If you have to buy a large, assuming n is large, then:

h1 + h2 + ... + hn = 1 * 48.61% +. (- 1) * (48.61% + 2.78%) = -0.0278

If you have to buy a small, empathy;

If any buy big buy small, but also empathy.

Thus, after n Bureau, the number of chips the player is: M * 97.22%

Visible If this continues, the number of each game bet or less certain, when the play a lot of innings, the players will only reduce the number of chips, only the principal amount of 97.22%, while 2.78% were washed away Makers the :(.

3, with experience of play:

1) The number of credits bet for x = N;

2) Buy the size of a plate out of the opposite;

3) If you win, continue to step 1), continue down if he loses;

4) The number of credits bet double x = 2 * x, proceed to step 2);

For this play, it seems earn no loss, but if bad luck once even opened a n a large, though this is a small probability that something will gamble in an instant, lose everything

In this case ignored bookmakers wash away 2.78%, can be wide open to open a small probability are regarded as 50%

N big / small probability even open to 1/2 ^ n, assuming that the purchase and use of chips, the number of coins bet for N * 2 ^ n, and the number lost to N * (1 + 2 ^ 1 + ...... + 2 ^ (n-1)) = N * (2 ^ n-1), when n is large that one can ignore, the number of chips left for MN * 2 ^ (n + 1) , that is, in the first innings will spend n N * 2 ^ (n + 1) of the funds, if insufficient funds remaining N * 2 ^ (n + 2), once lost the original capital inevitably difficult return.

N If you take no more than 10, N = 1000, then fired 10 large / small probability of 1/1024 less than 0.1%, while the required funds to about 2 million guaranteed not to gamble a blank. Although this looks like a very safe play In fact so each game is generally earn very little.

So you can bet in the end winning it? The answer is no, because every time to open a large open small is completely independent process, set P, no matter who buy big buy small bet, bet this event is set to Q, each open dice bet the whole process P * Q, or completely separate process, so when play many times, the number of chips the player is not increased, the dealer will be washed away 2.78%, makes money play is not present a.

Fourth, the model evaluation

By analyzing the mathematical methods, we find that play Sic Bo game, The winner is always the banker, 10 bet nine lost is the truth, for Gambling, Lottery is the same reason, it should not be too obsessed with, the sense of trying to do their job is success.

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