Expert: Scott A Wilson - 8/3/2010

Question
QUESTION: If I had a biased coin which landed 60:40 and could bet on it - what would be the ideal percentage of a pot to stake assuming a return of 1:1 per stake and full knowledge of the favoured side?

ANSWER: It would seem to me that to be 1/5 of the total cash would be a good amount to wager each time.
The cash held will never disappear and in the long hall, it should increase.


---------- FOLLOW-UP ----------

QUESTION: Hi Scott, thank you for your response. Please could you mathmatically / statistically justify your answer? I note 1/5 = 20% = the difference between the win/lose.. Were the biased coin 66.7/33.3 would the idealized stake be 1/3 (=33.3%) of the pot or were the coin 55:45 the stake should be 1/10 (=10%) of pot?

ANSWER: It just seemed like a good amount to me.
I calculated the expected amount of money given that the odds were 60:40.

This was made in Excel in a sheet called John.xls.

1.0   5.1598   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   $5.16
0.9   3.2519   1.0269   0.1441   0.0118   0.0006   0.0000   0.0000   0.0000   0.0000   0.0000   $4.44
0.8   1.9990   1.3327   0.3949   0.0682   0.0076   0.0006   0.0000   0.0000   0.0000   0.0000   $3.80
0.7   1.1951   1.2654   0.5955   0.1635   0.0288   0.0034   0.0003   0.0000   0.0000   0.0000   $3.25
0.6   0.6925   1.0388   0.6925   0.2693   0.0673   0.0112   0.0012   0.0001   0.0000   0.0000   $2.77
0.5   0.3874   0.7748   0.6887   0.3571   0.1190   0.0265   0.0039   0.0004   0.0000   0.0000   $2.36
0.4   0.2082   0.5354   0.6119   0.4079   0.1748   0.0500   0.0095   0.0012   0.0001   0.0000   $2.00
0.3   0.1069   0.3453   0.4958   0.4153   0.2236   0.0803   0.0192   0.0030   0.0003   0.0000   $1.69
0.2   0.0520   0.2080   0.3698   0.3835   0.2556   0.1136   0.0337   0.0064   0.0007   0.0000   $1.42
0.1   0.0238   0.1167   0.2545   0.3239   0.2650   0.1446   0.0526   0.0123   0.0017   0.0001   $1.20

The left most column is the amount you multiply what you have by what you bet.
In other words, the 1.0 means you risk it all; the 0.9 means you risk 90% every bet;
the 0.8 means you risk 80% every bet; the 0.7 means you risk 70% every bet, etc.

The right most column is the total expected value.

The 10 columns in between are for that many wins in the series of 10 bets, but in reverse.
Thus, the column with 5.1598 at the top is for winning it all; the next column is for 9 wins,
the next column is for 8 wins, the next column is for 7 wins, etc.

As you can see, the more that is wagerd, the more the first column has if there are 10 wins.
However, the chance of getting all 10 winds is 0.6^10, which is less than 1 in a hundred.
Note that if only a tenth of what you have is bet each time, the bottom row of the table should be looked at.  Here, even if you lose all 10 bets, you'll have 10,000th of the original money left.

The selection of 0.2 was only due to the fact that if 5 or more wins are gotten, you'll have more money than when you started.  This was only to ±0.05.  If 0.3 of the amount of moeny on hand is bet each time, you must get at least 6 wins to make a profit.  It is a larger profit on more wins, but how risky are you?

If we absolutely don't care about the money, bet it all every time.
Since you don't care about it, you'll have a big return if you win every time.
If a single loss would be unthinkable, you don't want to bet at all.


---------- FOLLOW-UP ----------

QUESTION: Hi Scott -please could you mail me a copy of the "john.xls" file - I'm interested to know what functions were applied to the individual boxes.
My reasons for this are that if I calculate 10 straight losses at 10% each of pot I find ~ 0.282 of original pot remaining, similarly for 10 straight wins I find my pot increased by a factor of ~ 2.428. These results are most disssimilar from yours - I suspect we have a misinterpretation of the original question

Answer
It would be good to have your address to send you the file.  Here it is in the proper format to be asted into a text file, though with that, the formulas aren't seen.  If I had you address, I could mail the Excel sheet as an attatchment.  It would actually have the calculations in it.

Now that I've though about it, it seems like the best way might be just to bet a constant amount each game.  Any loss of a large sum would really set the pereson back.

Odds of Game 0.6/0.4

Results after each bet is cell down below is a win, to the right and below is a loss.
wins=cells down from 1 at top - cells over from that column; losses=cells over from column 1.
1   1.5   0.5          
2   2.25   0.75   0.25          
3   3.375   1.125   0.375   0.125   
4   5.0625   1.6875   0.5625   0.1875   0.0625   
5   7.5938   2.5313   0.8438   0.2813   0.0938   0.0313   
6   11.3906   3.7969   1.2656   0.4219   0.1406   0.0469   0.0156
7   17.0859   5.6953   1.8984   0.6328   0.2109   0.0703   0.0234   0.0078   
8   25.6289   8.5430   2.8477   0.9492   0.3164   0.1055   0.0352   0.0117   0.0039   
9   38.4434   12.8145   4.2715   1.4238   0.4746   0.1582   0.0527   0.0176   0.0059   0.0020   
10   57.6650   19.2217   6.4072   2.1357   0.7119   0.2373   0.0791   0.0264   0.0088   0.0029   0.0010
         
Probability of each happening, the product of the last two.      
This table is the binomial; p=0.6, n=column A; calculated from two tables on right.
1   0.6   0.4          
2   0.36   0.48   0.16   
3   0.216   0.432   0.288   0.064
4   0.1296   0.3456   0.3456   0.1536   0.0256   
5   0.0778   0.2592   0.3456   0.2304   0.0768   0.0102   
6   0.0467   0.1866   0.3110   0.2765   0.1382   0.0369   0.0041
7   0.0280   0.1306   0.2613   0.2903   0.1935   0.0774   0.0172   0.0016
8   0.0168   0.0896   0.2090   0.2787   0.2322   0.1239   0.0413   0.0079   0.0007   
9   0.0101   0.0605   0.1612   0.2508   0.2508   0.1672   0.0743   0.0212   0.0035   0.0003   
10   0.0060   0.0403   0.1209   0.2150   0.2508   0.2007   0.1115   0.0425   0.0106   0.0016   0.0001

Expected amount of cash times the chance of that occurrence:
1   0.9   0.2          
2   0.81   0.36   0.04          
3   0.729   0.486   0.108   0.008          
4   0.6561   0.5832   0.1944   0.0288   0.0016          
5   0.5905   0.6561   0.2916   0.0648   0.0072   0.0003          
6   0.5314   0.7086   0.3937   0.1166   0.0194   0.0017   6.4E-05          
7   0.4783   0.7440   0.4960   0.1837   0.0408   0.0054   0.0004   1.3E-05         
8   0.4305   0.7653   0.5952   0.2645   0.0735   0.0131   0.0015   9.2E-05   2.6E-06      
9   0.3874   0.7748   0.6887   0.3571   0.1190   0.0265   0.0039   0.0004   2.1E-05   5.1E-07   
10   0.3487   0.7748   0.7748   0.4592   0.1786   0.0476   0.0088   0.0011   9.3E-05   4.6E-06   1.0E-07

Sum of the above rows is given by the fraction bet to the column A

Bet each time; changing this value to the values in the pay-off table.      

  10 wins   9 wins   8 wins   7 wins   6 wins   5 wins   4 wins   3 wins   2 wins   1 win   0 wins
1.0   6.1917   0   0   0   0   0   0   0   0   0   0
0.9   3.7072   1.3008   0.2054   0.0192   0.0012   5.0E-05   1.5E-06   2.9E-08   3.8E-10   3.0E-12   1.0E-14
0.8   2.1589   1.5992   0.5331   0.1053   0.0136   0.0012   7.5E-05   3.2E-06   8.8E-08   1.4E-09   1.1E-11
0.7   1.2190   1.4341   0.7592   0.2382   0.0490   0.0069   0.0007   4.6E-05   2.0E-06   5.3E-08   6.2E-10
0.6   0.6648   1.1081   0.8310   0.3694   0.1077   0.0215   0.0030   0.0003   1.8E-05   6.6E-07   1.1E-08
0.5   0.3487   0.7748   0.7748   0.4592   0.1786   0.0476   0.0088   0.0011   9.3E-05   4.6E-06   1.0E-07
0.4   0.1749   0.4997   0.6425   0.4895   0.2448   0.0839   0.0200   0.0033   0.0003   2.2E-05   6.3E-07
0.3   0.0834   0.2992   0.4834   0.4627   0.2907   0.1252   0.0375   0.0077   0.0010   8.3E-05   3.0E-06
0.2   0.0374   0.1664   0.3328   0.3944   0.3068   0.1636   0.0606   0.0154   0.0026   0.0003   1.1E-05
0.1   0.0157   0.0855   0.2100   0.3054   0.2915   0.1908   0.0867   0.0270   0.0055   0.0007   3.7E-05
         
Sum of rows, or the expected pay-off
6.191736422   , sum of columns B - L above
5.233835554   , sum of columns B - L above
4.411435079   , sum of columns B - L above
3.707221314   , sum of columns B - L above
3.105848208   , sum of columns B - L above
2.59374246   , sum of columns B - L above
2.158924997   , sum of columns B - L above
1.790847697   , sum of columns B - L above
1.480244285   , sum of columns B - L above
1.21899442   , sum of columns B - L above          

The number of wins would be when there are at most that many wins.
  10 wins   9 wins   8 wins   7 wins   6 wins   5 wins   4 wins   3 wins   2 wins   1 win   0 wins
1   6.1917   0   0   0   0   0   0   0   0   0   0
0.9   5.2338   1.5266   0.2258   0.0204   0.0012   0.0001   1.5E-06   3.0E-08   3.9E-10   3.0E-12   1.0E-14
0.8   4.4114   2.2525   0.6533   0.1202   0.0149   0.0013   0.0001   3.3E-06   9.0E-08   1.5E-09   1.1E-11
0.7   3.7072   2.4882   1.0541   0.2949   0.0567   0.0076   0.0007   4.8E-05   2.1E-06   5.3E-08   6.2E-10
0.6   3.1058   2.4410   1.3330   0.5019   0.1326   0.0248   0.0033   0.0003   1.8E-05   6.7E-07   1.1E-08
0.5   2.5937   2.2451   1.4702   0.6954   0.2362   0.0577   0.0100   0.0012   0.0001   4.7E-06   1.0E-07
0.4   2.1589   1.9840   1.4843   0.8418   0.3523   0.1075   0.0236   0.0036   0.0004   2.3E-05   6.3E-07
0.3   1.7908   1.7075   1.4083   0.9249   0.4622   0.1715   0.0463   0.0088   0.0011   0.0001   3.0E-06
0.2   1.4802   1.4428   1.2764   0.9436   0.5492   0.2424   0.0788   0.0182   0.0028   0.0003   1.1E-05
0.1   1.2190   1.2033   1.1178   0.9078   0.6024   0.3108   0.1200   0.0333   0.0062   0.0007   3.7E-05

         
ADDED:      
After looking at all of this complicated data, it might just be better to bet a small fixed amount each time.  If we had well over 20 and bet 1 each time, the results would be as follows.          
cash   10   8   6   4   2   0   -2   -4   -6   -8   -10
prob.   0.0060   0.0403   0.1209   0.2150   0.2508   0.2007   0.1115   0.0425   0.0106   0.0016   0.0001
change   0.0605   0.3225   0.7256   0.8600   0.5016   0.0000   -0.2230   -0.1699   -0.0637   -0.0126   -0.0010
         
The sum of the above row, which is the overall expected profit in betting 1 each time in 10 games: 2.

The following two table are used for computing the binomial distribution   
  0.6^(column P- columns over)*-.4^(columns over)          
1   0.6   0.4          
2   0.36   0.24   0.16   
3   0.216   0.144   0.096   0.064   
4   0.1296   0.0864   0.0576   0.0384   0.0256   
5   0.0778   0.0518   0.0346   0.0230   0.0154   0.0102   
6   0.0467   0.0311   0.0207   0.0138   0.0092   0.0061   0.0041
7   0.0280   0.0187   0.0124   0.0083   0.0055   0.0037   0.0025   0.0016   
8   0.0168   0.0112   0.0075   0.0050   0.0033   0.0022   0.0015   0.0010   0.0007
9   0.0101   0.0067   0.0045   0.0030   0.0020   0.0013   0.0009   0.0006   0.0004   0.0003

Sums would make no sense here, for the proper numer of each one has not been taken into account.

10   0.0060   0.0040   0.0027   0.0018   0.0012   0.0008   0.0005   0.0004   0.0002   0.0002   0.0001

Number of ways each could happen:
1   1   1   1          
2   1   2   1          
3   1   3   3   1          
4   1   4   6   4   1          
5   1   5   10   10   5   1          
6   1   6   15   20   15   6   1          
7   1   7   21   35   35   21   7   1         
8   1   8   28   56   70   56   28   8   1      
9   1   9   36   84   126   126   84   36   9   1   
10   1   10   45   120   210   252   210   120   45   10   1
         
Note that the sum of each row above is 2^n.