Quote:
Originally Posted by 18000rpm
Let's say the odds of winning in a single game is P, where P is between 0 and 1 (eg. 0.42 for Blackjack, which has roughly 42% chance of winning a hand).
How do you calculate the probability you will be ahead after N games? Assuming the same bet each game and an endless bankroll?
There are many ways to find the probability of a net win after N trials and not just 1.
IF you are looking to use Blackjack, one can to use a formula by D. Schlesinger from Blackjack Attack (I use it in Excel) that uses variance and normal distribution or it can be found in a calculator over at qfit.com.
What game exactly are you looking for or does it have to be a game?
another said this
"To summarize, what you seek is a simple sum of binomial probs.
This is typically available as the complement of an expression given in many spreadsheets and programming languages."
Just 1 method that works for some games and a very incomplete answer, imo.
One can use the normal distribution, simulations and transition matrix type solutions just to name a few more methods, not to mention simulations.
You really need to be more specific as to the type of win and game and IF exact probabilities are required or not.
The below answers using a binomial probability solution assume
equal $$$ bets every time, same probability of winning each time
and only a win or lose result
and enough $$$ to make each and every wager.
Here, a few examples I did real fast using a program
as doing this by long hand, for me, is a waste of time and very boring.
Code:
100 trials each:
probabilities of a NET WIN
using binomial probability distribution
*****
even money bets payoff
00 Roulette
0.2650235 (51+ wins)
1k trials: 0.0447959 (501+ wins)
even money payoff
0 Roulette
0.3553499 (51+ wins)
1k trials: 0.1876306 (501+ wins)
<<<<< >>>>>
2 to 1 bets payoff
00 Roulette
0.3357928 (34+ wins)
1k trials: 0.1144736 (334+ wins)
2 to 1 payoff
0 Roulette
0.4052573 (34+ wins)
1k trials: 0.2668903 (334+ wins)
<<<<< >>>>>
1 number bet
00 Roulette 35 to 1 payoff
0.4915671 (3+ wins)
1k trials: 0.3961392 (28+ wins)
0 Roulette 35 to 1 payoff
0.5093944 (3+ wins)
1k trials: 0.4511495 (28+ wins)
*****
pass line craps - no odds - even money payoff
0.4045094 (51+ wins)
1k trials: 0.3159887 (501+ wins)
pass line craps - 345X odds
0.48240218 (using transition matrix)
simulation results (1 million, 100 game sessions)
Bankroll increased . . = 48.243% of the time
1k trials: 0.4618313 (using transition matrix)
simulation results (1 million, 1000 game sessions)
Bankroll increased . . = 46.167% of the time
<<<<< >>>>>
Hard 8 (or Hard 6) 9 to 1 payoff
0.2994398(11+ wins)
1k trials: 0.1460037(101+ wins)
closer look here:
even money bets payoff
00 Roulette
0.2650235 (51+ wins)
51 wins is N/2 +1 because 50 wins and 50 losses results on a 0 net gain.
you asked for "the probability you will be
ahead after N games?"
1k trials: 0.0447959 (501+ wins)
using a calculator here:
binomial probability calculatorilities
enter:
n=100
k=51
p=18/38
result:
P: 51 or more out of 100
0.265023450545
****
2 to 1 bets payoff
00 Roulette
0.3357928 (34+ wins)
100/3 = 33 wins. that still results in a net loss.(66 + -67 = -1)
N/3 +1 = 34 wins. (68 + -66 = 2)
back to the calculator
n=100
k=34
p=12/38
result:
P: 34 or more out of 100
0.335792839527
Both results in agreement with my results from my program(s) I use.
<<<<< >>>>>
I can go on without possibly answering your question, so I ask you to be way more specific.
As one can see the pass line bet at Craps with odds is even more fun to calculate.
Oh yes, I forgot that with blackjack, one can also take the result distribution (like in video poker) and use a transition matrix to calculate the result, as one can win 1 , 1.5 or 2 or more units each round as well as pushing and losing 1 or 2 units each round. way more challenging to calculate. many just simulate many sessions of play to get a good approximation.
Computers do well computing if the program is written properly, imo.
I also like to know the probability of being 'down' after N trials
because
ending up 'even'
after N trials
sometimes 'feels' like a 'win'
hope this helps more than it hurts
Last edited by kraps2312; 01-20-2018 at 12:05 PM.
Reason: link messed up