What Is the Rate of Rise of The Gambler's Money In 22Bet Id?
(1/n)*log(W_{n}/W_{0}) = log[(1 + f*(d-1))^{k}(1 – f)^{(n-k)}]^{1/n}
The gross return for a win in 22Bet Id is written in blue, and the gross return for a loss is written in red. When a refund is given, the account amount stays the same, so W1 = W0 and W1/W0 = 1. If you had k wins, m returns, and n-k-m losses after n bets, this expression could be changed to:
(1/n)*log(W_{n}/W_{0}) = log[(1 + f*(d-1))^{k}(1 – f)^{(n-k-m)}(1)^{m}]^{1/n}
The number 1 multiplied by any power is always 1, so this phrase can be shortened to:
(1/n)*log(W_{n}/W_{0}) = log[(1 + f*(d-1))^{k}(1 – f)^{(n-k-m)}]^{1/n}
Note that when n is big enough, (n-k-m)/n = q. Put the above formula into the steps in Part 2 and solve for f to get:
f=p(d-1)-q/(p+q)(d-1)
When q + p = 1, this formula becomes the standard Kelly criterion, which we talked about in Parts 1 and 2. Also, keep in mind that the part in the denominator (p+q) could be written as (1-j), where j is the chance of getting a refund.
The decimal odds method, which is popular in Australia, is used in the above formula. For those who are more familiar with the "b to 1" odds used in other books, the appropriate formula is:
f=pb-q/(p+q)b
Interpretation
The effect of a possible return depends on whether it makes it less likely to win or less likely to lose, or both. When betting on football on bet365, the 0-0 draw refund doesn't change the chance of winning, but it does make it less likely that you'll lose, since the refund only applies to lost bets.
In tennis betting, if the bookmaker cancels and returns all bets if a player gets hurt, the odds of both winning and losing are lower because either player could quit hurt. Still, it's not unusual for a player to get hurt during a game, but they still keep playing. You might think that the person who is hurt is more likely to leave the game than the other player. This would change the way you think about your chances of winning and losing in different ways. Another thing to think about is that if a player is hurt and still playing, they may be more likely to quit if they are losing than if they are winning.
The Arsenal vs. Chelsea game was looked at again.
Remember the made-up match between Arsenal and Chelsea of 22Bet Id? To sum up, here are the odds, with your estimated chances of each happening in brackets:
Full-time result:
Arsenal: 2.60 (p = 20% chance that it will happen)
Chelsea: 2.65 (p = chance of happening 50% of the time)
Draw: 3,25 (p = 30% chance that it will happen)
Using The Standard Kelly Criterion, You Should Bet
f = [0.5(2.65 – 1) – 0.50]/(2.65 – 1) = 0.197 = 19.7% of your account amount on Chelsea.
Now, let's say the bookmaker pays back all lost full-time bets when the score is 0-0. If you think there's a 5% chance of a 0-0 draw, and you think the odds of the following outcomes now:
- Arsenal: 2.60 P = 20% Chance of winning, q = 75% chance of losing, j = 5% chance of getting money back
- Chelsea: 2.65 P = 50% Chance of winning, q = 45% chance of losing, j = 5% chance of getting money back
- Draw: 3.25 P = 30% Chance of winning, q = 70% chance of losing, j = 0% chance of getting money back
Using the Kelly criterion and taking into account the chance of getting your money back, you should bet f = [0.5(2.65–1)–0.45]/[(2.65–1)(0.5+0.45)]. = 0.239 = 23.9% of the money in your Chelsea account.
In this case, the optimal bet size has grown because the estimated chance of losing has gone down.
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