Hi guys. Please tell what you think about this. Is it correct? Is it wrong? Why? All comments are much appriciated. Criticism also welcomed. I posted same in Probability forum, but I am not getting any answers, so I hope you guys can maybe help me. Thanks.
COIN FLIP GAME
Rules of a game:
- Flipping a coin
- We do not have infinity amount of units to bet
- Our lowest vs highest bet is 20x
- We decide about our next pick:H or T, after last one falls
- If we do make correct prediction for next H/T we get +0,999 unit
- If we do not make correct prediction for next T/H we lose 1 unit
- Infinity number of flips
First some calculations:
Number of trials:4
Number of combinations:2 on 4=16
All possible combinations( H: Heads; T: Tails)
1.HHHH
2.THHH
3. HTHH
4. HHTH
5. HHHT
6. TTTT
7. HTTT
8. THTT
9. TTHT
10. TTTH
11. HHTT
12. HTHT
13. HTTH
14. THTH
15. TTHH
16. THHT
We bet all the combinations that has as a result 2H+2T, HHHH and TTTT. There are 8 combinations like this, mentioned above (combinations with numbers: 1,6,11,12,13,14,15,16).
How do we bet:
We repeat that to infinity. If we lose before 4th bet, we wait until 5th bet and repeat as mentioned.
More calculations:
N=4
u= 4*(0,5)=2; 0,5 because it is coin flip; u is mean
o2=4*(0,5)*(1,0-0,5)=1; 0,5 because it is coin flip
o=square root 1=1
Let us now take 0,5o and 1,5o.
0,5o=0,382924; rounded on 6th decimal; not in favour of this theory
1,5o=0,866386; rounded on 6th decimal; not in favour of this theory
1-0,866386=0,133614
We are interested in u+-0,5o. This in our case means from 1,5 to 2,5 (2 is mean).
We are also interested in +-1,5o. This in our case means from 0,5 to 3,5 (2 is mean).
If we would have odds 2 for every winning prediction we would need to hit our system in at least:
winning combinations/all combinations
8/16=0,5=50 % of the time.
The numbers from SD calculations tells us that we will hit this system in: 38,2924% + 13,3614 %=51,6538 % of time.
51,6538 % is more than 50 %.
Therefor, we made a +EV bet out of only 0EV bets.
Now, let us calculate EV in our case from the example (with odds 1,999).
In our case we have odds 1,999.
That means that every of our 8 combinations has got odds:
1,999 on 4=15,968; rounded on 3 decimal; not in favour of this system
Together our system has got odds: 15.968
We calculated before, that we will win our system in 51.6538 % of time.
15,968 x 0,516538=8,2480; rounded on 4 decimal;not in favour of our system;
8,2480 is more than 8.
In every 4 flips we make profit of 8,2480 - 8= 0,248 units.
That is profit per flip: 0,248/4=0,062 .
Profit per every flip =0,062/8=0,755%.
The more time we play this game the more units we have (if we are betting like I wrote).
Therefor, we made a +EV bet out of only -EV bets.
If above calculation is correct. This means that those 2 sentences are wrong:
- The EV of the sum of random variables is always equal to the sum of the EVs, whether the variables are independent or not (for example in the same game).
- Optimal strategy is allways the best strategy.