Probability from 500 to 6500 in BlackJack

[FreddyBoy3333]

I would like to know how to calculate the odds of reaching 6500 from 500. I am going to flat bet 500 non stop until I reach 6500. The House Edge will be 0,5% exluding that I am not able to double up or split when I am at 500. What are the odds of reaching 6500 and how many bets does it take avarage? Is this something you can calculate or do I have to simulate this?

Any replys very much appreciated
Thank you

[FreddyBoy3333]

no one here willing to reply my post? I've seen BruceZ beeing good at this kind of math. Could you help me out here?

[BruceZ]

[ QUOTE ]
I would like to know how to calculate the odds of reaching 6500 from 500. I am going to flat bet 500 non stop until I reach 6500. The House Edge will be 0,5% excluding that I am not able to double up or split when I am at 500. What are the odds of reaching 6500 and how many bets does it take average? Is this something you can calculate or do I have to simulate this?

Any replys very much appreciated
Thank you

[/ QUOTE ]

This is interesting because I get almost the same answer of about 7.5% by two different methods that use completely different models. The first method uses only the 0.5% house advantage and a standard deviation of 1.16 bets for 1 hand which is typical of a 6-deck game. The second method uses an approximate model of blackjack which assigns a hand loss percentage of 48%, push 9%, non-pushed blackjack 4.5%, winning double or pair split 2%, and single bet win 36.5%. These final 3 values give a probability of winning of 43%. Blackjacks are assumed to pay 3-2. I am ignoring the change in these numbers due to not being able to split or double when the bankroll has 1 bet.

Method 1

Consider that the house is playing with an EV per hand of 0.005 bets, SD of 1.16 bets, and bankroll of 12 bets. The house's risk of ruin if it played forever (not stopping when it wins your 1 bet) would be

ror = exp(-2*0.005*12/1.16^2) =~ 0.914681329

Now let r be the house's chance of going broke before it wins a single bet. Then we have

ror = r + (1-r)*ror^(13/12)

This says that to go broke playing forever, the house can either go broke before it wins a single bet with probability r, or it can win a single bet with probability 1-r, and then lose a 13 unit bankroll with probability ror^(13/12). Solving this for r gives

r = [ror - ror^(13/12)] / [1 - ror^(13/12)]

Substituting the value for ror computed above gives

r =~ 7.4%.

Since this is the probability of the house losing a 12 bet bankroll before it wins a single bet, this is the same as the probability that the player wins 12 bets before he loses a single bet.

Method 2

This time let ror represent the player's risk of ruin if he plays forever (not stopping after winning 12 bets). The model of blackjack described above results in the following equation:

ror = 0.48 + 0.09*ror + 0.365*ror^2 + 0.045^ror^2.5 + 0.02^ror^3

This comes from the same idea as the formula above. That is, we can lose the first hand with probability 0.48, or we can push with probability 9% and maintain the same ror, or we can win a single bet with probability 36.5% and then lose a 2 unit bankroll with probability ror^2, etc. Using the Excel goal seek function gives

ror =~ 0.999785931.

Then if we let r denote the probability that we go broke before our bankroll reaches 13 bets, we then using the same idea as above, we have the following equation relating r and ror:

ror = r + (1-r)*ror^13

Solving for r gives

r = (ror - ror^13) / (1 - ror^13)

Substituting the value for ror found above gives r =~ 0.922978073. Since this is the probability that we go broke before we win 13 bets, the probability that we succeed in winning the 13 bets is

1 - r =~ 7.7%.


Refer to the links below for more information about the risk of ruin formula used in method 1 above.

Bankroll Formulas

Derivation of Bankroll Formulas

Online calculators for bankroll and risk of ruin

[pzhon]

[ QUOTE ]
no one here willing to reply my post? I've seen BruceZ beeing good at this kind of math. Could you help me out here?

[/ QUOTE ]
These types of problems are simple if you are playing something that pays off an integer number of bets. It's extremely complicated and messy when you take into account that blackjacks pay 3:2. What is your plan if you get down to $250 and can't bet $500?

Anyway, that's why when people ask this question for blackjack, as they have many times in the past, no one is enthusiastic about answering. Generally, the person asking hasn't realized that their question is messy and underspecified, and often the person would be as satisfied with the answer to a much simpler question which doesn't involve blackjack.