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## Probability-Related Question

Hi, I work as an Actuary and love statistics (and blackjack). There's a blackjack-related problem that I've been trying to wrap my head around, but I can't quite figure it out.

To make the math easier (even though they're not perfectly accurate), here are a few assumptions:

-Assume that the probability of winning a hand of blackjack is .5.
-Also assume that all bets placed are the same size (1 unit bets).
-An example of "going up" 2 hands in any given series would be WLWW or WW or LWWW (wins and losses cancel out).
-Similarly, an example of "going down" 3 hands in a given series would be LLL or WLWLLLL or LWLLL.

Question: In any given series of hands dealt, what is the probability of going up 5 hands before going down 10 hands? Or, a similar question, what's the probability of going up 3 hands before going down 5 hands? This is similar to a geometric distribution but the solution is slightly different. I'm looking for a formula that can be applied to any scenario of "going up" x hands before "going down" y hands.

Any thoughts?

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You need to factor in the probability of tie. Further, actual win probability is overstated, which obviously translates to actual loss probability being understated, which means that your formula create skewed results.

Not withstanding the above, what is your purpose?

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-Ties are disregarded (i.e. if the series was WWLT, you would disregard the tie, the series would revert to WWL, and you'd continue on).

-Yes, assuming no card counting and basic strategy, win probability is slightly overstated and the results will skew slightly more favorable than the true probability.

My purpose is to calculate ballpark probabilities for various scenarios before playing. I currently do not count cards, so the longer I play the more I'm likely to lose. I would like to be able to walk into a game knowing there's a 70% chance (for example) that I can win x hands before losing y hands.

I understand that this 70% probability would be slightly less because of my assumptions, but I am comfortable making those assumptions for these purposes.

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You need to determine firstly if you plan on playing perfect basic, or if your preferred approach is looking like an idiot. Assuming the former, which will assist your win factor, you can reliably, long term, expect to win 43% of hands, lose 47% of hands. Regardless of your philosophy, you must factor in ties, as they will also play a factor in splits and doubles.

I’m not sure how to reliably give you the answer you are seeking. However, the following parameters should be included - you are odds on (99% of the time - would need to look at an average sim to come out with a closer answer) favored to lose the next hand. Regardless of any win streak encountered, playing basic will guarantee long term loss, regardless of any short term success.

Last but not least, no disrespect intended - why not expend the effort on a project that will assist you in reversing a losing game into a winning game.

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Originally Posted by mp13
Hi, I work as an Actuary and love statistics (and blackjack). There's a blackjack-related problem that I've been trying to wrap my head around, but I can't quite figure it out.

To make the math easier (even though they're not perfectly accurate), here are a few assumptions:

-Assume that the probability of winning a hand of blackjack is .5.
-Also assume that all bets placed are the same size (1 unit bets).
-An example of "going up" 2 hands in any given series would be WLWW or WW or LWWW (wins and losses cancel out).
-Similarly, an example of "going down" 3 hands in a given series would be LLL or WLWLLLL or LWLLL.

Question: In any given series of hands dealt, what is the probability of going up 5 hands before going down 10 hands? Or, a similar question, what's the probability of going up 3 hands before going down 5 hands? This is similar to a geometric distribution but the solution is slightly different. I'm looking for a formula that can be applied to any scenario of "going up" x hands before "going down" y hands.

Any thoughts?
Using your parameters, and excluding ties, the answers are 66.67% and 62.5%, respectively. I'm sure you see where the two values come from: 10/10+5 and 5/3+5, respectively.

Again, your 0.5 win rate is overstating reality. 0.48 is more like it, excluding ties.

Hope this helps.

Don

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